aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Stabilities of Parallel Flowand Horizontal Convection
Liang SUN
Report of Postdoc Research
Academic Advisor Prof. Yun-Fei Fu
LSRSCE, School of Earth and Space Sciences,University of Science and Technology of China,November, 2007 bstract
In the first part, the stability of two-dimensional parallel flow is discussed. A more restrictivelygeneral stability criterion for inviscid parallel flow is obtained analytically. First, a sufficient criterionfor stability is found as either − µ < U ′′ U − U s < < U ′′ U − U s in the flow, where U s is the velocityat the inflection point, and µ is the eigenvalue of Poincar´e’s problem. Second, this criterion is gen-eralized to barotropic geophysical flows in the β plane. Based on the stability criteria, the necessarycondition for wave-mean flow interaction is also obtained.Then, the general stability criteria of two-dimensional inviscid rotating flow with angular velocityΩ( r ) are obtained analytically. First, a necessary instability criterion for centrifugal flows is derivedas ξ ′ (Ω − Ω s ) < ξ ′ / (Ω − Ω s ) < ξ ′ is the vortictiy of profileand Ω s is the angular velocity at the inflection point ξ ′ = 0 . Second, a criterion for stability is foundas − ( µ + 1 /r ) < f ( r ) = ξ ′ Ω − Ω s < µ is an eigenvalue. The new criteria are the analoguesof the criteria for parallel flows, which are special cases of Arnol’d’s nonlinear criteria. Specifically,Pedley’s cirterion is proved to be an special case of Rayleigh’s criterion. Moreover, the criteria forparallel flows can also be derived from those for the rotating flows. The analogy between rotation andstratification in inviscid flow is also addressed. These results extend the previous theorems and wouldintrigue future research on the mechanism of hydrodynamic instability.Besides, the essence of shear instability is fully revealed within the linear context. The mechanismof shear instability is explored by combining the mechanisms of both the Kelvin-Helmholtz instability(K-H instability) and resonance of waves. The shear instability requires both a concentrated vortex(with speed of U s ) in the flow and resonant waves to interact with the concentrated vortex. Physically,the standing waves (with phase speed c r = U s ) can interact with the concentrated vortex, so theycan trigger instability via K-H instability in the flows. While the travelling waves (with c r = U s )have no interaction with the concentrated vortex, so that they can not trigger instability in the flows.The resonance of waves are totally within the linear context.In consequence, the above criteria would be helpful for understanding the wave-mean flow in-teraction, especially the Rossby wave-mean flow interaction in barotropic flows. According to thestable criteria, the necessary condition for wave-mean flow interaction can be obtained. And why thedisturbed waves can’t take energy from the mean flow in the stable flow is revealed. If the flow isstable, there is no wave-mean flow interaction at all. This explains why the disturbed waves can’ttake energy from the mean flow in the stable flow.In the second part, we report the numerical simulations of the partial-penetrating flow in hor-izontal convection within a squire cavity tank at high Rayleigh numbers 10 < Ra < . Thepartial-penetrating flow was first reported in the experiment by Wang and Huang (2005), which isthough of an important material to understand ocean circulation energy budget. The fast establishedbut slowly steadied flow is simulated, where a shallow and closed circulation cell is obtained numer-3 bstract ically as partial-penetrating flow for the first time, which is consistent with the experiment. As thepartial-penetrating flow is shallow, it is seldom affected by the bottom boundary. The depth of partial-penetrating circulation satisfies minus 1/5-power law of Rayleigh number. The larger the Rayleighnumber is, the shallow the partial-penetrating flow is. An objective definition of partial-penetratingis given based on this power law. Then, further investigation points out that the Prandtl numbergoverns the partial-penetrating flows. As Pr ≥ ≤ < Ra < , three continues regimes are obtained: linear regime ( 10 < Ra < ), transition regime( 10 < Ra < ) and 1/5-power law regime ( 10 < Ra < ). For the flow strength, a 1/3-powerlaw of Ra is fitted when Ra is not high enough ( 10 < Ra < ). However, a 1/5-power law isobtained as Ra is high enough ( 10 < Ra < ). The 1/5-power law confirms Rossby’s analysisand implies that 1/3-power law of Ra for Nusselt number by Siggers et al. is over estimation.Finally, the critical Rayleigh number for unset of the horizontal convection is also addressed. Theflow is found to be unsteady at high Rayleigh numbers. There is a Hopf bifurcation of Ra from steadysolutions to periodic solutions, and the critical Rayleigh number Ra c is obtained as Ra c = 5 . × for the middle plume forcing at P r = 1 , which is much larger than the formerly obtained value.Besides, the unstable perturbations are always generated from the central jet, which implies that theonset of instability is due to velocity shear (shear instability) other than thermally dynamics (thermalinstability). Finally, Paparella and Young’s (2000) second hypotheses about the destabilization of theflow is numerically proved, i.e. the middle plume forcing can lead to a destabilization of the flow.The report was supported by the National Foundation of Natural Science (No. 40705027) andthe National Science Foundation for Post-doctoral Scientists of China (No. 20070410213).4 ontents
Abstract 31 General Stability Criteria 1 § § § § § § § § § § § § § § § § § § § § § § § § ONTENTS § § § § § § § § § § § § § Reference 51 ist of Figures − ≤ y ≤ ξ = cos( πy/
2) , dashed)is neutrally stable, while profile 1 ( ξ = cos( y ) , solid) and profile 3 ( ξ = cos(2 y ) , dashdoted) are stable and unstable, respectively. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. 51-3 Growth rate ω i as an function of wavenumber k , (a) for U = tanh(1 . y ) , (b) for U = tanh(1 . y ) , both within the interval − ≤ y ≤ U ( y ) and vorticity ξ ( y ) , right ones depict the disturbance of vorticities. Theunstable veloctiy profile U ( y ) has a local maximum in vorticity ξ ( y ) (a). If the vortices(A, B and C) disturbed from their original positions (dashed line) to new places (solidcurve), they will be taken away from their original positions due to pressure difference.The stable veloctiy profile U ( y ) has a local minimum in vorticity ξ ( y ) (b). Thedisturbed vortices (A’, B’ and C’) will be brought back to their original positions dueto pressure difference. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. 151-6 Angular velocity Ω (solid line) and corresponding vorticity ξ (dashed line) versus r . . 202-1 Snapshots of the flow fields (streamfunction Ψ ), with solid counter curves for Ψ > < > < > < Ra = 5 × , withsolid counter curves for Ψ > < > E k ( a ) and maximum flowΨ max ( b ) at P r = 8 and Ra = 5 × . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. 307 IST OF FIGURES max ( a ) and the heat flux N u ( b ) vs. Rayleigh number,where Ra p = RaP r / in ( a ). ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 312-9 ( a ) The maximum of streamfunction Ψ max vs time t for Ra = 2 × . The solid,dashed and dash-doted curves are solutions with N = 40 , N = 64 and N = 80 ,respectively. ( b ) The stable and unstable regime on the plot of Rayleigh number Ra vs N . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 342-10 The flow fields (streamfunction Ψ ) at four different time steps: t = 0 . a ), t = 1 . b ), t = 1 . c ) and t = 2 . d ) at Pr=8 and Ra = 5 × , solid curves for clockwiseflow ( Ψ > < >
20 ) are shadowed and the counter intervals are 10 forΨ > A = 1 ( a ), A = 0 . b ) and A = 0 .
25 ( c ) at Pr=8 and Ra = 5 × . Thepartial-penetrating cells are shadowed and the counter intervals are 10 in each figures. .. 372-12 The flow fields (streamfunction Ψ ) for full-penetrating flow at Pr=1 and Ra = 5 × .The counter interval is 5, and Ψ >
35 is shadowed. ... ... ... ... ... ... ... ... ... ... ... ... ... ... 382-13 The flow fields (streamfunction Ψ ) near the forcing surface of three respective aspectratios: A = 1 ( a ), A = 0 . b ) and A = 0 .
25 ( c ) at Pr=1 and Ra = 5 × . Thestreamfunction Ψ >
35 are shadowed and the counter intervals are 5 in each figures. .... 382-14 D c vs. Ra. The solid, dashed and dash dotted lines are power laws of Ra respectivelyfor D c at Pr = 6 , Pr = 4 and Pr = 2 . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 392-15 The flow stream function (a) and temperature field (b) of Ra = 10 . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positiveand negative values, respectively. . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 402-16 The flow stream function (a) and temperature field (b) of Ra = 10 . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positiveand negative values, respectively. . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 412-17 The flow stream function Ψ max (a) and heat flux (b) vs. Ra. ... ... ... ... ... ... ... ... ... ... 422-18 The flow stream function (a) and temperature field (b) of Ra = 5 × . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positiveand negative values, respectively. . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 452-19 The horizontal (a) and vertical (b) velocity fields of Ra = 5 × . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positiveand negative values, respectively. . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 452-20 Growth rate σ r (solid) and σ i (dashed) vs. Ra , respectively. .. ... ... ... ... ... ... ... ... ... 462-21 The perturbational vorticity fields at t = 0 (a), t = T / t = T / t = 3 T / Ra = 5 . × , solid and dashed curves for positive and negative values,respectively. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 472-22 The perturbational vorticity fields at t = T / t = 5 T / t = 6 T / t = 7 T / Ra = 5 . × , solid and dashed curves for positive and negative values,respectively. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 472-23 The vorticity of Ra = 5 . × with vertical velocity w (shadowed as w > IST OF FIGURES Ra = 5 × , which are steady andstable and symmetric with sidewall plume forcing. . ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. 489 IST OF FIGURES hapter 1 1 General Stability Criteria § § The instability due to shear in the flow is one of the fundamental and the most attracting problemsin many fields, such as fluid dynamics, astrophysical fluid dynamics, oceanography, meteorology, etc.More generally, shear instability is also referred as barotropic instability in geophysical flows, wherethe gravitational and buoyancy effects are ignored. Shear instability has been intensively investigated,which is to the greatly helpful understanding of other instability mechanisms in complex shear flows.Rayleigh investigated the growth of linear disturbances by means of normal mode expansion, whichleads to Rayleigh’s equation [1]. Using this equation, Rayleigh first proved a necessary criterion forinstability, i.e., Inflection Point Theorem, which is also called Rayleigh-Kuo theorem [e.g. 2] for Kuo’sgeneralization to barotropic geophysical flows in the β plane [3]. Then, Fjortoft found a strongernecessary criterion for instability [4]. Besides, Tollmien gave a heuristic result that the criteria arealso sufficient for instability if the velocity profiles are the symmetrical or monotonic [5]. These criteriaare well known and have been widely used in various applications [e.g. 2, 6, 7].On the other hand, Arnol’d considered the shear instability in a totally different way [8, 9, 10]. Hestudied the conservation law of the inviscid flow via Euler’s equations and found two nonlinear stabilitytheorems by means of variational principle. Arnol’d’s first stability theorem corresponds to Fjørtoft’scriterion [6, 11]. However, Arnol’d’s second nonlinear stability theorem, has no such correspondinglinear criterion. Though Arnol’d’s second nonlinear theorem is more useful in the geophysical flows[11], is seldom known by the scientists in other fields. Dowling suggested that Arnol’d’s idea shouldneed to be added to the general fluid-dynamics curriculum [11]. Yet his suggestion has not beenfollowed until now [e.g. 2, 6, 7, 12], since the proofs of Arnol’d’s theorems are very advanced andcomplex in mathematics.The aim of this section is to find the elementary proofs for Arnol’d’s theorems, which could beused to teach undergraduate students. As the variational method is too advanced and complex forundergraduate students. The new proofs are obtained in a totally different way, where the linearstability problem is considered by using Rayleigh’s equation.1 hapter 1 General Stability Criteria § To find the criteria, Rayleigh’s equation for an inviscid parallel flow is employed [1, 2, 6, 7, 13],which is the vorticity equation of the disturbance [6, 7]. For a parallel flow with mean velocity U ( y )in Fig.1-1, the vorticity is conserved along pathlines. The amplitude of disturbed flow streamfunction,namely φ , satisfies( φ ′′ − k φ ) − U ′′ U − c φ = 0 , (1-1)where k is the nonnegative real wavenumber and c = c r + ic i is the complex phase speed and doubleprime ′′ denotes d /dy . The real part of complex phase speed c r is the wave phase speed, and ω i = kc i is the growth rate of the wave. This equation is to be solved subject to homogeneousboundary conditions φ = 0 at y = a, b. (1-2)There are three main categories of boundaries: (i) open channels with both a and b being finite, (ii)boundary layers with either a or b being infinite, and (iii) free shear flows with both a and b beinginfinite.It is obvious that the criterion for stability is ω i = 0 ( c i = 0 ), for that the complex conjugatequantities φ ∗ and c ∗ are also a physical solution of Eq.(1-1) and Eq.(1-2).Multiplying Eq.(1-1) by the complex conjugate φ ∗ and integrating over the domain a ≤ y ≤ b ,we get the following equations Z ba [( k φ ′ k + k k φ k ) + U ′′ ( U − c r ) k U − c k k φ k ] dy = 0 , (1-3)and c i Z ba U ′′ k U − c k k φ k dy = 0 . (1-4)Rayleigh used only Eq.(1-4) to prove his theorem, i.e., a necessary condition for instability is U ′′ ( y s ) =0 , where y s is the inflection point and U s = U ( y s ) is the velocity at y s . Fjørtoft noted that Eq.(1-3)should also be satisfied, then he obtained his necessary criterion. To find a more restrictive criterion,we shall investigate the conditions for c i = 0 . Unlike the former investigations, we consider this y xU(y) ab Figure 1-1: Sketch of parallel flow.2 problem in a totally different way: if the velocity profile is stable ( c i = 0 ), then the hypothesis c i = 0should result in contradictions in some cases. Following this, some more restrictive criteria can beobtained.To find a stronger criterion, we need to estimate the ratio of R ba k φ ′ k dy to R ba k φ k dy . This isknown as Poincar´e’s problem: Z ba k φ ′ k dy = µ Z ba k φ k dy, (1-5)where the eigenvalue µ is positive definite for any φ = 0 . The smallest eigenvalue value, namely µ ,can be estimated as µ > ( πb − a ) , like what Tollmien have done [5] .Then using Poincar´e’s relation Eq.(1-5), a new stability criterion may be found: the parallel flowis stable if − µ < U ′′ U − U s < f ( y ) = U ′′ U − U s , where f ( y ) is finite atthe inflection point. We will prove the criterion by two steps. First, we will prove proposition 1: ifthe velocity profile is subject to − µ < f ( y ) < c r = U s .Proof: Since − µ < f ( y ) < − µ < U ′′ U − U s = U ′′ ( U − U s )( U − U s ) ≤ U ′′ ( U − U s )( U − U s ) + c i . (1-6)Substitution of c r = U s and Eq.(1-6) into Eq.(1-3) results in Z ba [ k φ ′ k + k k φ k + U ′′ ( U − U s ) k U − c k k φ k ] dy > . (1-7)This contradicts Eq.(1-3). So proposition 1 is proved.Then, we will prove proposition 2: if − µ < f ( y ) < c r = U s , there must be c i = 0 .Proof: If c i = 0 , then multiplying Eq.(1-4) by ( c r − U t ) /c i , where the arbitrary real constant U t does not depend on y , and adding the result to Eq.(1-3), yields Z ba [( k φ ′ k + k k φ k ) + U ′′ ( U − U t ) k U − c k k φ k ] dy = 0 . (1-8)But the above Eq.(1-8) does not hold for some special U t . For example, if U t = 2 c r − U s , then thereis ( U − U s )( U − U t ) < k U − c k , and U ′′ ( U − U t ) k U − c k = f ( y ) ( U − U s )( U − U t ) k U − c k > − µ . (1-9)This yields Z ba {k φ ′ k + [ k + U ′′ ( U − U t ) k U − c k ] k φ k } dy > , (1-10)which also contradicts Eq.(1-8). So proposition 2 is also proved.Using “proposition 1: if − µ < f ( y ) < c r = U s ” and “proposition 2: if − µ < f ( y ) < c r = U s then c i = 0 ”, we find a stability criterion. If the velocity profile satisfies − µ < U ′′ U − U s < < f ( y ) , which isequivalent to Fjørtoft’s criterion. Thus we have the following theorem.3 hapter 1 General Stability Criteria Theorem 1: If the velocity profile satisfies either − µ < U ′′ U − U s < < U ′′ U − U s , the flow isstable.This criterion covers Rayleigh’s and Fjørtoft’s criteria. And the proofs here are very simple andeasy to understand comparing to Arnol’d’s proofs. As mentioned above, we have investigated thestable criterion via Rayleigh’s equation, while Arnol’d [14] considered the hydrodynamic stability ina totally different way. Is there any relationship between these proofs? Two points are outlined here.First, this criterion is essentially the same as Arnol’d’s second stability theorem and is more restrictivethan Fjørtoft’s criterion. Second, the proofs here are similar to Arnol’d’s variational principle method.For the arbitrary real number U t , which is like a Lagrange multiplier in variational principle method,plays a key role in the proofs. § One may note that the criterion above is something different from Fjørtoft’s criterion. Why arethe functions of U ′′ / ( U − U s ) used in Arnol’d’s theorems and present theorems, unlike U ′′ ( U − U s ) inFjørtoft’s theorem? This is due to the property of Rayleigh’s equation. It can be seen from Eq.(1-1)that the stability of profile U ( y ) is not only Galilean invariant, but also independent from the themagnitude of U ( y ) due to the linearity. So the stability of U ( y ) is the same as that of AU ( y ) + B ,where A and B are any arbitrary nonzero real numbers. As the value of U ′′ ( U − U s ) is only Galileaninvariant but not magnitude free, it satisfies only part of the Rayleigh’s equation’s properties. On theother hand the value of U ′′ / ( U − U s ) satisfies both conditions, this is the reason why the criteria inboth Arnol’d’s theorems and present theorems are the functions of U ′′ / ( U − U s ) . Since the stabilityof inviscid parallel flow depends only on the velocity profile’s geometry shape, namely f ( y ) , and themagnitude of the velocity profile can be free, then the instability of inviscid parallel flow could be called”geometry shape instability” of the velocity profile. This distinguishes from the viscous instabilityassociated with the magnitude of the velocity profile.Moreover, the above theorem is essentially associated with vorticity distribution in the flow field.As known from Fjørtoft’s criterion, the necessary condition for instability is that the base vorticityprofile ξ = − U ′ has a local maximum. Note that U ′′ / ( U − U s ) ≈ ξ ′′ s /ξ s near the inflection point,where ξ s is the vorticity at the inflection point, that means that the base vorticity ξ must be convexenough near the local maximum for instability, i.e., the vorticity should be concentrated somewherein the flow for instability. Otherwise, the flow is stable if the vorticity distribution is smooth enoughnear the inflection point at y s . A simple example can be obtained by following Tollmien’s way [5].Fig.1-2 depicts three vorticity profiles within the interval − ≤ y ≤ y = 0 . Profile 2 ( U = − πy/ /π ) is neutrally stable, while profile 1 ( U = − sin( y ) ) and profile3 ( U = − sin(2 y ) / U = tanh( αy ) within the interval − ≤ y ≤ α is a constant. This velocity profile is anclassical model of mixing layer, and has been investigated by many researchers (see [2, 7, 13] andreferences therein). Since U ′′ ( U − U s ) = − α tanh ( αy ) / cosh ( αy ) < − ≤ y ≤ α according to both Rayleigh’s and Fjørtoft’s criteria. But it can be derived fromTheorem 1 that the flow is stable for α < π / α = 1 . α = 1 . U ( y ) and U ( y ) . The growth rate of the profiles can be obtained by Chebyshevspectral collocation method [13] with 100 collocation points, as shown in Fig.1-3. It is obvious that c i = 0 for U and c i > U , which agrees well with the criteria obtained above. This is also4 -0.25 0 0.25 0.5 0.75 1-1-0.75-0.5-0.2500.250.50.751 y ξ Figure 1-2: Vorticity profiles within the interval − ≤ y ≤ ξ = cos( πy/
2) , dashed) isneutrally stable, while profile 1 ( ξ = cos( y ) , solid) and profile 3 ( ξ = cos(2 y ) , dash doted) are stableand unstable, respectively.a counterexample that Fjørtoft’s criterion is not sufficient for instability. So this new criterion forstability is more useful in real applications.Then, recalling the proof of theorem 1, we will find that the following Rayleigh’s quotient I ( f )plays a key role in determining the stability of parallel flows. I ( f ) = min φ R ba [ k φ ′ k + f ( y ) k φ k ] dy R ba k φ k (1-11)Note that the proof of theorem 1 is still valid in the case of I ( f ) > I ( f ) > I ( f ) . Theorem 1 ismore convenient for the real applications in different research fields.The present stability criteria give a affirmative answer to the questions at the beginning, i.e.,there are some stable flows if U ′′ ( U − U s ) < f ( y ) > k -14 -13 -12 -11 -10 (a) ω i k (b) -3 -3 -3 -2 ω i Figure 1-3: Growth rate ω i as an function of wavenumber k , (a) for U = tanh(1 . y ) , (b) for U = tanh(1 . y ) , both within the interval − ≤ y ≤ hapter 1 General Stability Criteria U s ’’=0U’’>0 or U’’<0 f(y)>0f(y)>- µ f(y)<- µ stable stablestableunstableA BCD Figure 1-4: Diagram of stable velocity profiles.criterion), and (iii) − µ < f ( y ) < f ( y ) < − µ and f ( y ) changing sign within the interval. However, if f ( y ) changes sign somewhere within the interval [ a, b ] , then the parallel flow is stable. For that f ( y )changing sign implies U ′′′ s = 0 but U ′′′′ s = 0 , so U ′′ does not change sign near the inflection point.Thus c i must vanish in Eq.(1-4), i.e., the parallel flow is stable for f ( y ) changing sign within theinterval. In this way, the parallel flow might be unstable only for f ( y ) < µ somewhere, which willintrigue further studies on this problem. In fact, there are still stable flows if µ < f ( y ) is violated.Finally, the stable criterion for the parallel inviscid flows can be applied to the barotropic geo-physical flows in a differentially rotating system. Considering the barotropic plane flows in a rotatingframe, which are the approximations of barotropic geophysical flows [2, 3, 11], Eq.(1-1) changes to( φ ′′ − k φ ) − U ′′ − βU − c φ = 0 , (1-12)where β is the gradient of the Coriolis parameter with respect to y . Eq.(1-12) is a generalizedRayleigh’s equation, and there is a generalized stability criterion for these flows.Theorem 2: The flow is stable, if the velocity profile satisfies either − µ < U ′′ − βU − U s < < U ′′ − βU − U s , where U s is the velocity at the point U ′′ ( y s ) = β .The above criteria would be helpful for understanding the wave-mean flow interaction, especiallythe Rossby wave-mean flow interaction in barotropic flows. According to the stable criteria, thenecessary condition for wave-mean flow interaction can be obtained. And why the disturbed wavescan’t take energy from the mean flow in the stable flow is revealed. If the flow is stable, there is nowave-mean flow interaction at all.First, when the velocity profile has no inflection point, then the speeds of barotropic waves c r − U = 0 . According to Eq.(1-3), U − c r > U ′′ < U if the vortex gradient is positive. This extendsthe west-propagation theory of Rossby waves. And U − c r < U ′′ > U if the vortex gradient is negative.Second, when the velocity profile has an inflection point U ′′ s = 0 , the speed of most favorite wavewhich might have interaction with the mean flow should be c r = U s . However, the wave speed c r = U s holds for − µ < U ′′ U − U s < β plane. Thus, the waves have no interaction with the stableflows. This is the reason why the disturbed waves can’t take energy from the stably mean flow.6 On the other hand, [15] pointed out that c r of a unstable wave must lie between the minimumand the maximum values of the mean velocity profile. Thus, the unstable wave is stationary relativeto the mean flow, and can take energy from the mean flow. So there are wave-mean flow interactionsin unstable flows. § In summary, the general stability criteria are obtained for inviscid parallel flow. Both the criteriaand the proofs are remarkably simple and easy to understand, comparing to Arnol’d’s nonlineartheorems. The new criteria extend the former theorems proved by Rayleigh, Tollmien and Fjørtoft.This may shed light on the flow control and investigation of the vortex dynamics. Based on thestability criteria, the reason why the disturbed waves can’t take energy from the stably mean flow isexplained that there is no wave-mean flow interaction at all. § § Shear instability, caused by the velocity shear, is one of most important instabilities in the flows.Although the mechanism of shear instability is not full revealed yet, it is applied to explain theinstability of mixing layer, jets in pipe, wakes behind cylinder, etc. Some simple models were employedto study the shear instability, including Kelvin-Helmholtz (K-H) model, piecewise linear velocity profile[16] and continued velocity profiles [1], etc. To reveal the mechanism of shear instability, the stabilityof perturbation waves should be understood. In K-H model, the growth rate of the disturbance waveis proportion to the product of wavenumber and velocity shear, thus the shortwaves are more unstablethan longwaves in K-H model. However, Rayleigh [16] found that the piecewise linear profile is linearlyunstable only in a finite range of wavenumbers 0 ≤ k < k c , which means the shortwaves are stablein this case [2, 7, 16, 17]. A contradiction [2, 7] emerges as shortwave and longwave dominate theshear instability in K-H model and piecewise linear profiles, respectively. This was explained either asviscous effect must be considered for shortwave [2] or as longwaves do not ”feel” the finite thicknessof the layer [2, 7].To dispel the contradiction, we should find out which dominates the shear instability, shortwaveor longwave? It is from the numerical simulation (see [2, 7, 13] and references therein) and sometheoretical analysis (e.g. Tollmien [5] and Lin [18]) that longwave might be more unstable thanshortwave. They also proved the long-wave instability subject to the velocity profile U is eithersymmetric or monotone. We will investigate this problem following the way by Sun [19].Moreover, a prior estimation of growth rate, which may be obtained from the investigation, isuseful for unstable flows. For example, Howard’s semicircle theorem [15] has been used to validatethe numerical calculations [13]. Another useful result was found by Høiland [20] and Howard [15] thatthe growth rate ω i must less equate to half of the maxim of vorticity, i.e., ω i ≤ | U ′ | max / U ′ is always great than zero even thevelocity profile has no inflection point. So this estimation is trivial for these cases. Here we will showa refinement estimation of growth rate, which can be applied to general velocity profiles in parallelflows. 7 hapter 1 General Stability Criteria The motivation of this short letter is to investigate these problems within the context of inviscidparallel flow. The aim here is to find out some general characters for unstable waves. § To this purpose, long-wave instability in shear flows is investigated via Rayleigh’s equation [1, 2,7, 17]. For a parallel flow with mean velocity U ( y ) , where y is the cross-stream coordinate. Thestreamfunction of the disturbance expands as series of waves (normal modes) with real wavenumber k and complex frequency ω = ω r + iω i , where ω i relates to the grow rate of the waves. The flow isunstable if and only if ω i > φ , holds( φ ′′ − k φ ) − U ′′ U − c φ = 0 , (1-13)where c = ω/k = c r + ic i is the complex phase speed. The real part of complex phase speed c r = ω r /k is the wave phase speed. This equation is to be solved subject to homogeneous boundary conditions φ = 0 at y = a, b. (1-14)From Rayleigh’s equation, we get the following equations: Z ba [( k φ ′ k + k k φ k ) + U ′′ ( U − c r ) k U − c k k φ k ] dy = 0 , (1-15)and c i Z ba U ′′ k U − c k k φ k dy = 0 . (1-16)Before the further discussion, we need estimate the rate of R ba k φ ′ k dy to R ba k φ k dy , as Sun did[19]. This is known as Poincar´e’s problem: Z ba k φ ′ k dy = µ Z ba k φ k dy, (1-17)where the eigenvalue µ is positive definition for φ = 0 . The smallest eigenvalue value, namely µ ,can be estimated as µ > ( πb − a ) . And an auxiliary function f ( y ) = U ′′ U − U s is also introduced, where f ( y ) is finite at inflection point.With the preparation above, we have such consequence. If − f ( y ) < Q < ∞ , where Q is apositive constant, then the disturbances with shortwaves k > k c are always stable, where k c is acritical wavenumber subject to k c = Q − µ . We will prove the consequence by two steps. At first,we prove proposition 1: if c r = U s , the disturbances with shortwaves k > k c are always stable.Proof: Since U ′′ = f ( y )( U − U s ) and c r = U s , this yields to U ′′ ( U − U s ) k U − c k > f ( y ) U ′′ U − U s > − Q, (1-18)8 and Z ba [( k φ ′ k + k k φ k ) + U ′′ ( U − U s ) k U − c k k φ k ] dy ≥ Z ba [( µ + k c + U ′′ ( U − U s ) k U − c k ) k φ k ] > . (1-19)This contradicts Eq.(1-15). So proposition 1 is proved.Then, we prove proposition 2: if c r = U s , there must be c i = 0 with k > k c .Proof: Otherwise if c i = 0 , so according to Eq.(1-15) and Eq.(1-16), for any arbitrary realnumber U t which does not depend on y , it holds Z ba [( k φ ′ k + k k φ k ) + U ′′ ( U − U t ) k U − c k k φ k ] dy = 0 . (1-20)But the above Eq.(1-20) can not be hold for some special U t . For example, let U t = 2 c r − U s , thenthere is ( U − U s )( U − U t ) < k U − c k , and U ′′ ( U − U t ) k U − c k = f ( y ) ( U − U s )( U − U t ) k U − c k > − Q. (1-21)For k > k c , this yields to Z ba {k φ ′ k + [ k + U ′′ ( U − U t ) k U − c k ] k φ k } dy > , (1-22)which also contradicts Eq.(1-20). So proposition 2 is also proved. These two propositions are naturalgeneralization of stabile criterion proved by Sun [19].From the above two propositions, we can draw a conclusion that the disturbances with shortwaves k > k c are always stable. This means that the shear instability in flows must be long-wave instability.Furthermore, the short-wave stability means that without any viscous effect, the shortwaves can alsobe damped by shear flow. This mechanism is unlike the viscous mechanism that the viscosity hasa damping effect on especially the shortwaves. It implies that the shear flow itself can damp theshortwaves. This is very general and important conclusion, which explains why the instabilities foundin shear flows are mostly long-wave instabilities. § This result is also very important for numerical calculation, which means shortwaves can betruncated in the calculations without changes the stability of shear flow. So the growth rates oflongwaves have enough information for judging the stability of shear flow. On the other hand, thetruncation of longwaves would probably change the instability of the shear flow. So the streamwiselength scale must be longer enough to have longwaves for the numerical simulations in shear flows,such as plane parallel flow and pipe flow. Otherwise the instability of shear flow would be dampedwithout long-wave perturbations.Then the growth rate of unstable waves can be estimated here by following the former investiga-tion. Similar to the assumption above, − f ( y ) < Q = p µ < ∞ , where 1 < p < ω i ≤ ( p − √ µ | U max − U min | .9 hapter 1 General Stability Criteria Proof: It is from Eq.(1-15) that gives Z ba k k φ k dy = Z ba − [ k φ ′ k + U ′′ ( U − c r ) k U − c k k φ k ] dy. (1-23)Substituting Eq.(1-17) to Eq.(1-23) and recalling that µ < µ , this yields Z ba k k φ k dy ≤ Z ba − [ U ′′ ( U − c r ) k U − c k + µ ] k φ k dy. (1-24)Multiplying above inequality (1-24) by c i , we get ω i Z ba k φ k dy ≤ Z ba h ( y ) k φ k dy, (1-25)where h ( y ) = − [ U ′′ ( U − c r ) k U − c k + µ ] c i . (1-26)Suppose the maxim of h ( y ) is P , then the growth rate is subject to ω i ≤ P . (1-27)This follows h ( y ) ≤ − [ U ′′ ( U − c r )( U − c r ) + c i + µ ] c i . (1-28)Substitution of f ( y ) into Eq.(1-28), gives h ( y ) ≤ µ [ ( p − U − c r ) − c i ( U − c r ) + c i ] c i . (1-29)When c i = ( p − U − c r ) , (1-30)the right hand of Eq.(1-29) get its largest value P = ( p − µ ( U − c r ) . (1-31)Then the growth rate must be subject to ω i ≤ ( p − √ µ | U max − U min | , (1-32)where U min and U max are minimum and maximum of U ( y ) , respectively. And the wavenumber k max corresponding to the largest growth rate is k max = p ( p − µ . (1-33)So the result is proved. One should note that the fast growth rate ω i is only an approximation, butnot a precision one, so as to wavenumber k max .Comparing with the previous one by Høiland and Howard, the above estimation includes notonly a more precision estimation about the the growth rate of unstable waves but also the regime ofunstable wavenumbers. This would be much helpful for validation in numerical calculations.As well known, the instability due to velocity shear is always associated to Kelvin-Helmholtzinstability (K-H instability), in which the disturbances of all wavelengths are amplified. According10 to K-H instability, the shorter the wavelength is, the faster the perturbation wave amplifies. As theabove investigation points out that the shortwaves are always more unstable than longwaves in thiscase, the contradiction at the beginning can be dispelled.An physical explanation [2, 7] to the contradiction is that the K-H instability model has nointrinsic length scale, while Rayleigh’s model has width of shear layer as length scale of waves. Thiscan be noted from that Rayleigh’s case reduces to the Kelvein-Helmholtz vortex sheet model in thelong-wave limit k ≪ k max is proportion to √ µ , then thecorresponding wave length λ max is approximately 2( b − a ) / √ p − ω i ≤ ( p − √ µ ( U max − U min ) . This estimation extend the previous result obtained byHøiland and Howard. Both results are important in numerical applications. The first one providesthe estimation of unstable wavenumbers, and the second one provides the estimation of growth rateof unstable waves. These results may be useful on both numerical calculation and stability analysis. § § The hydrodynamic instability is a fundamental problem in many fields, such as fluid dynamics,astrodynamics, oceanography, meteorology, etc. There are many kinds of hydrodynamic instabilities,e.g., shear instability due to velocity shear, thermal instability due to heating, viscous instabilitydue to viscosity and centrifugal instability due to rotation, etc. Among them, the shear instability(inviscid) is the most important and the simplest one, which has been intensively explored (see [2, 6, 7]and references therein). Both linear and nonlinear stabilities of shear flow have been considered, andsome important conclusions have also be obtained from the investigations.On the one hand, the nonlinear stability of shear flow has been investigated via variationalprinciples. Kelvin [21] and Arnol’d [9, 10, 14] have developed variational principles for two-dimensionalinviscid flow [22]. They showed that the steady flows are the stationary solutions of the energy H .And if the second variation δ H is definite, then the steady flow is nonlinearly stable. Moreover,Arnol’d proved two nonlinear stability criteria, and that the flow is linearly stable as δ H is positivedefinite [10, 14, 22, 23]. However, Arnol’d’s theorem is not convenient to use as δ H is alwaysindefinite in sign, except for two special cases (see [23] and references therein). Due to the lack of theexplicit expressions in both H and the stability criteria the variational principle is inconvenient forreal applications. 11 hapter 1 General Stability Criteria On the other hand, the linear stability of shear flow has also been investigated via Rayleigh’sequation. Within the linear context, there are three important general stability criteria, namelyRayleigh’s criterion [1], Fjørtoft’s criterion [4] and Sun’s criterion [19]. As all the criteria have explicitexpressions, they are more convenient in real applications, and are widely used in many fields. Basedon the previous investigations, Sun [19] also pointed out that the flow is stable for Rayleigh’s quotient I ( f ) > H have not been revealed before,although it is very important in variational principles. The connection between linear and nonlinearstability criteria should be retrieved explicitly. The relationships between instability mechanisms andthe physical explanation for shear instability are needed. The aim of this paper is to reveal the essenceof the shear instability by investigating the inviscid shear flows in a channel. And this would lead toa more comprehensive understanding on shear instabilities. § For the two-dimensional inviscid flows with the velocity of U , the vorticity ξ = ∇× U is conservedalong pathlines [7, 22, 24]: dξdt = ∂ξ∂t + ( U · ∇ ) ξ = 0 . (1-34)Its linear disturbance reduces to Rayleigh’s equation provided the basic flow U being parallel. Con-siders an shear flow with parallel horizontal velocity U ( y ) in a channel, as shown in Fig.1-5. Theamplitude of disturbed flow streamfunction ψ , namely φ , satisfies [2, 6, 7] :( φ ′′ − k φ ) − U ′′ U − c φ = 0 , (1-35)where k is the nonnegative real wavenumber and c = c r + ic i is the complex phase speed and doubleprime ′′ denotes the second derivative with respect to y . The real part c r is the phase speed ofwave, and c i = 0 denotes instability. This equation is to be solved subject to homogeneous boundaryconditions φ = 0 at y = a, b .Considers that the velocity profile U ( y ) has an inflection point y s at which U ′′ s = U ′′ ( y s ) = 0and U s = U ( y s ) . As Sun [19] has pointed out, the following Rayleigh’s quotient I ( f ) > flow is stable. I ( f ) = min φ R ba [ k φ ′ k + f ( y ) k φ k ] dy R ba k φ k dy (1-36)where f ( y ) = U ′′ U − U s . While if I ( f ) < k N = − I ( f ) and c r = U s . Moreover, there are unstable modes with c r = U s and c i = 0 if I ( f ) < ≤ k < k N .This can also be proved by following the way by Tollmien [5], Friedrichs [6, 29, 30] and Lin [18]. Thus I ( f ) = 0 means the flow is neutrally stable. And there is only one neutral mode with k = 0 and c r = U s in the flow. These conclusions can be summarized as a new theorem.Tollmien-Fridrichs-Lin theorem: The flows are neutrally stable, if I ( f ) = 0 . The flows are stableand unstable for I ( f ) > I ( f ) < k ≫ k N ) are always more stablethan longwaves (e.g. k ≪ k N ) in the inviscid shear flows [30, 31]. So the shear instability is due tolong-wave instability, and the disturbances of shortwaves can be damped by the shear itself withoutany viscosity [31].As mentioned above, the linear stability criterion can be derived from the nonlinear one [14].Moreover, the nonlinear stability criterion can also be obtained from the linear one. To illuminatethis, the nonlinear criterion is retrieved explicitly via the above theorem, which is briefly proved asfollows.Similar to Arnol’d’s definition, the general energy H here is defined as H = 12 Z ba [ 12 k∇ Ψ k + h (Ψ)] dy (1-37)where Ψ( y ) is the streamfunction of the flow with ∂ Ψ ∂y = U ( y ) − U s , and h (Ψ) is a function of Ψ .The variation of δH = 0 gives △ Ψ = h ′ (Ψ) . (1-38)So Arnol’d’s variational principle is retrieved. And the function h = k∇ Ψ k / δ H holds δ H = 14 Z ba [ k∇ ψ k + h ′′ (Ψ) ψ ] dy, (1-39)where ψ denotes the variation δ Ψ . Noting that h ′′ (Ψ) remains unknown, Arnol’d’s nonlinear criteriahave not be extensively used. Fortunately, we can obtain the explicit expression of h ′′ (Ψ) here viathe investigation on the linear stability criterion. For ∂ Ψ ∂y = U − U s , h ′′ (Ψ) = dh ′ d Ψ = dh ′ dy dyd Ψ = U ′′ ( U − U s ) = f ( y ) . (1-40)As h ′′ (Ψ) is solved explicitly, δ H has an explicit expression, which is greatly helpful for real appli-cations.Let the streamfunction of the perturbation expands as travelling waves ψ ( x, y, t ) = φ ( y ) e i ( kx − ωt ) ,where ω is the frequency. The averages of k∇ ψ k and k ψ k are ( k φ ′ k + k k φ k ) / k φ k / x , respectively. So Eq.(1-39) reduces to δ H = 18 Z ba [ k φ ′ k + k k φ k + f ( y ) k φ k ] dy. (1-41)13 hapter 1 General Stability Criteria The sign of δ H is then associated with I ( f ) in Eq.(1-36). If I ( f ) < δ H can be both negative and positive, i.e., the stationary solution is a saddle point. And I ( f ) > δ H is positive definite and vice versa. So the stable flow has the minimum value of the totalkinetic energy H . The physical meaning of H can also be revealed, as the explicit expressions of H , ∂ Ψ ∂y = U − U s and h ′′ (Ψ) = f ( y ) have been obtained.First, according to the expressions, the velocity U in vorticity conservation law Eq.(1-34) can bedecomposed to two parts: the rotational flow U − U s and the irrotational advection flow U s . Thevorticity ξ in Eq.(1-34) depends only on U − U s , and U s is only advection velocity. Then U − U s and U s are associated with the dynamics and kinetics of the flow, respectively. Eq.(1-34) physically showsthat the vorticity field is advected by U s , which can also known from the conservation of vorticity inthe inviscid flows. A similar example is the dynamics of vortex in the wake behind cylinder, where thevortices dominate the dynamics of the flow and they are advected by mean flow (see Fig.2 in [32]).The decomposition of velocity may be useful in vortex dynamics, for that our investigation clearlyshows that the dynamics of the flow is dominated by vorticity distribution.Then the physical meaning of H can also be understood from the above investigation. It is not U but U − U s that is associated with the general energy H , so H = R ba ( U − U s ) dy is not thetotal kinetic energy but the kinetic energy of flow with vorticity. Thus the stable steady states arealways minimizing the kinetic energy of the flow associated with vorticity. This is also the reason whythe flow with maximum vorticity might be unstable, as Fjørtoft’s criterion shows. We would like torestate it as a theorem and to name it after Kelvin [21] and Arnol’d [14] for their contributions onthis field [22].Kelvin-Arnol’d theorem: the stable flow minimizes the kinetic energy of flow associated withvorticity.Both Tollmien-Fridrichs-Lin theorem and Kelvin-Arnol’d theorem are equivalent to the followingsimple principle [33]: The flow is stable, if and only if all the disturbances with c r = U s are neutrallystable. § We have obtained the sufficient and necessary conditions for instability, then the physical mech-anism of instability can be understood from them. According to Fjørtoft’s and Sun’s criteria [19], thenecessary conditions for instability require that the base vorticity must be concentrated enough (e.g.sheet vortex). We call it ”concentrated vortex” for latter convenience. The following investigationwill reveal that the essence of shear instability is due to the interaction between the ”concentratedvortex” and the corresponding resonant waves.As mentioned above, the concentrated vortex is a general model of sheet vortex in the Kelvin-Helmholtz model, for that the sheet vortex can be recovered as the concentrated vortex ξ ( y s ) → ∞ (see[12] for a comprehensive discussion about the Kelvin-Helmholtz model and continued shear profiles).Then how the shear flow becomes unstable, if there is a concentrated vortex? As the sufficientcondition for instability is I ( f ) < ≤ k < k N with c r = U s are unstable. These modes are stationary or standing waves, comparing to the velocity at inflectionpoint. So that the shear instability is due to the disturbance of concentrated vortex by the standingwaves with c r = U s . In this case, the resonance mechanism is valid. The interaction waves propagateat the same speed with the concentrated vortex, so that they are locked together and amplified bysimple advection [27]. In short, the disturbances on concentrated vortex is amplified like that in14 Kelvin-Helmholtz model.This instability mechanism combines both K-H instability and resonance mechanism. Physically,the standing waves (with c r = U s ) can interact with the concentrated vortex, so they can triggerinstability in the flows. While the travelling waves (with c r = U s ) have no interaction with theconcentrated vortex, so that they can not trigger instability in the flows. This is the mechanism ofshear instability. As pointed out above, the inviscid shear instability is due to long-wave instability.If the longwaves are unstable, they can obtain the energy from background flows. Thus, the energywithin small scales transfer to and concentrate on large scales. In a word, the shear instability itselfprovides a mechanism to inverse energy cascade and to maintain the large structures or coherentstructures in the complex flows.To illuminate the mechanism of shear instability, a physical diagram is also presented here byfollowing the way of interpreting the K-H instability [24]. Fig.1-5 sketches the mechanism of shearinstability in terms of wave disturbances of vortices. The mean velocity profile U ( y ) has an inflectionpoint at y s = 0 with U s = 0 , and the corresponding vorticity is ξ ( y ) = − U ′ ( y ) . There is a localmaximum at y s in the unstable vorticity profile in Fig.1-5a. According to Eq.(1-34), the vorticity isconserved in the inviscid flows. If the vortices at the local maximum (A, B and C) are sinusoidallydisturbed from their original positions (dashed line) to new places (solid curve), they have negativevorticities with respect to the undisturbed ones. The vortices will induce cyclone flows around themin consequence. The flows around the vortices become faster (slower) in the upper (lower) of vortex Areferred to the basic flow U ( y ) . The pressures at upper and lower decrease (indicated by - signs) andincrease (indicated by + signs) according to Bernoulli’s theorem, respectively. Then vortex A getsa upward acceleration due to the disturbed pressure difference as the uparrow shows. This tends totake the vortex away from its original position, so the flow is unstable. On the other hand, Fig.1-5bdepicts the disturbances in a stable velocity profile, where a local minimum is in the vorticity profile.The disturbed vortices have positive vorticities with respect to the undisturbed ones. The vortices U(y) ξ (y) -+ y - -+ + A B C(a) U(y) ξ (y) +- y + +- - A’ B’ C’(b) Figure 1-5: Sketch of shear instability: physical interpretation. Left parts depict the profiles of velocity U ( y ) and vorticity ξ ( y ) , right ones depict the disturbance of vorticities. The unstable veloctiy profile U ( y ) has a local maximum in vorticity ξ ( y ) (a). If the vortices (A, B and C) disturbed from theiroriginal positions (dashed line) to new places (solid curve), they will be taken away from their originalpositions due to pressure difference. The stable veloctiy profile U ( y ) has a local minimum in vorticity ξ ( y ) (b). The disturbed vortices (A’, B’ and C’) will be brought back to their original positions dueto pressure difference. 15 hapter 1 General Stability Criteria will induce anticyclone flows around them in consequence. So vortex A’ get a downward accelerationdue to the disturbed pressure difference as the downarrow shows. This tends to bring the vortexback from its original position, so the flow is stable. In this interpretation, the advection of U s isindependent of the shear instability, only the flow field of U − U s and the corresponding vorticity ξ are the dominations. The unstable disturbances in Fig. 1-5 have c r = U s , which consists with theabove discussions. This physically explains why the maximum and minimum vorticities have differentstable aspects.Though Tollmien-Fridrichs-Lin theorem is a sufficient and necessary condition for stability, theunknown φ in the theorem restricts its application. So some simple criteria may be more useful. Forthe parallel flows within interval a ≤ y ≤ b , there are two simple criteria.Corollary 1: The flow is stable for f ( y ) > − ( πb − a ) [19, 33].Corollary 2: The flow is unstable for f ( y ) < − ( πb − a ) [33].In summary, the general stability and instability criteria are retrieved for inviscid parallel flowwithin linear context, which are associated with the nonlinear stability criteria, i.e., minimizing thekinetic energy of flow. Then the mechanism of shear instability is explained as the resonance ofstanding waves with the concentrated vortex at c r = U s . The physical process is also sketched byextending the way of interpreting K-H instability. Finally, some useful criteria are given. These resultswould lead to a more comprehensive understanding on shear instabilities, especially for undergraduatestudents. § § The instability of the rotating flows is one of the most attractive problems in many fields, suchas fluid dynamics, astrophysical hydrodynamics, oceanography, meteorology, etc. Among them, thesimplest one is the instability of pure rotation flow between coaxial cylinders, i.e., Rayleigh-Taylorproblem, which has intensively been explored [6, 17].Two kinds of instabilities in inviscid rotating flow have been theoretically addressed in the liter-atures. One is centrifugal instability, which was first investigated by Rayleigh [1, 6]. He derived thecirculation criterion for the inviscid rotating flows that a necessary and sufficient condition for stabilityto axisymmetric disturbances is that the square of the circulation does not decrease anywhere. Thiscriterion is also be stated as the Rayleigh discriminant Φ ≥ Motivated, then, by the theoretical criteria for parallel flows [4, 19], our study focuses on theinstability due to shear in inviscid rotating flows. The aim of this letter is to obtain such criteria forthe inviscid rotating flows, and the relationship between previous criteria is also discussed. Thus otherinstabilities may be understood via the investigation here. § For this purpose, Howard-Gupta equation (hereafter H-G equation) [36] is employed. To obtainH-G equation, Euler’s equations [2, 6, 17, 24] for incompressible barotropic flow in cylindrical polarcoordinates ( r, θ ) are then given by ∂u r ∂t + u r ∂u r ∂r + u θ r ∂u r ∂θ − u θ r = − ρ ∂p∂r , (1-42)and ∂u θ ∂t + u r ∂u θ ∂r + u θ r ∂u θ ∂θ + u r u θ r = − ρr ∂p∂θ . (1-43)Under the condition of incompressible barotropic flow, the evolution equation for the vorticity can beobtained from Eq.(1-42) and Eq.(1-43), ∂ξ∂t + u r ∂ξ∂r + u θ r ∂ξ∂θ = 0 , (1-44)where ξ = r ∂∂r ( ru θ ) − r ∂u r ∂θ is the vorticity of the background flow. Eq.(1-44) can also be derivedfrom Fridman’s vortex dynamics equation [22, 24]. And it admits a steady basic solution, u r = 0 , u θ = V ( r ) = Ω( r ) r, (1-45)where Ω( r ) is the mean angular velocity. And Rayleigh discriminant is defined byΦ = 1 r ddr (Ω r ) . (1-46) § Then, consider the evolution of two-dimensional disturbances. The disturbances ψ ′ ( r, θ, t ) , whichdepend only on r , θ and t , expand as series of waves, ψ ′ ( r, θ, t ) = φ ( r ) e i ( nθ − ωt ) , (1-47)where φ ( r ) is the amplitude of disturbance, n is real wavenumber and ω = ω r + iω i is complexfrequency. Unlike the wavenumber in Rayleigh’s equation for inviscid parallel flows, the wavenumber n here must be integer for the periodic condition of θ . The flow is unstable if and only if ω i > φ satisfies( n Ω − ω )[ D ∗ D − n r ] φ − nr ( Dξ ) φ = 0 , (1-48)where D = d/dr , D ∗ = d/dr + 1 /r . This equation is known as H-G equation and to be solved subjectto homogeneous boundary conditions Dφ = 0 at r = r , r . (1-49)17 hapter 1 General Stability Criteria By multiplying rφ ∗ ω − Ω n to H-G equation Eq.(1-48), where φ ∗ is the complex conjugate of φ , andintegrating over the domain r ≤ r ≤ r , we get the following equation Z r r r { φ ∗ ( D ∗ D ) φ − [ n r + nD ( ξ ) r ( n Ω − ω ) ] k φ k } dr = 0 . (1-50)Then the integration gives Z r r r {k φ ′ k + [ n r + n (Ω n − ω ∗ ) ξ ′ r k Ω n − ω k ] k φ k } dr = 0 , (1-51)where φ ′ = Dφ , ξ ′ = D ( ξ ) and ω ∗ is the complex conjugate of ω . Thus the real part and imagepart are Z r r r {k φ ′ k + [ n r + (Ω − c r ) ξ ′ r k Ω − c k ] k φ k } dr = 0 , (1-52)and Z r r c i ξ ′ k Ω − c k k φ k dr = 0 , (1-53)where c = ω/n = c r + ic i is the complex angular phase speed. Rayleigh used only Eq.(1-53) to provehis theorem: The necessary condition for instability is that the gradient of the basic vorticity ξ ′ mustchange sign at least once in the interval r < r < r . The point at r = r s is called the inflectionpoint with ξ ′ s = 0 , at which the angular velocity of Ω s = Ω( r s ) . This theorem is the analogue ofRayleigh’s inflection point theorem for parallel flow [1, 6].Similar to the proof of Fjørtoft theorem [4] in the parallel flow, we can prove the followingcriterion.Theorem 1: A necessary condition for instability is that ξ ′ (Ω − Ω s ) < ξ ′ / (Ω − Ω s ) < r ) is stable ( c i = 0 ), then the hypothesis c i = 0 should result incontradictions in some cases. So that a more restrictive criterion can be obtained.To find the criterion, we need estimate the rate of R r r r k φ ′ k dr to R r r k φ k dr , Z r r r k φ ′ k dr = µ Z r r k φ k dr, (1-54)where the eigenvalue µ is positive definition for φ = 0 . According to boundary condition Eq.(1-49), φ can expand as Fourier series. So the smallest eigenvalue value, namely µ , can be estimated as µ > r π / ( r − r ) .Then there is a criterion for stability using relation (1-54), a new stability criterion may be found:the flow is stable if − ( µ + 1 /r ) < ξ ′ Ω − Ω s < f ( r ) = ξ ′ Ω − Ω s , where f ( r ) is finite atinflection point. We will prove the criterion by two steps. At first, we prove proposition 1: if thevelocity profile is subject to − ( µ + 1 /r ) < f ( r ) < c r = Ω s .18 Proof: Since − ( µ + 1 /r ) < f ( r ) < − ( µ + 1 /r ) < ξ ′ Ω − Ω s ≤ ξ ′ (Ω − Ω s )(Ω − Ω s ) + c i , (1-55)and if c r = Ω s and 1 ≤ n , this yields to Z r r r k φ ′ k + [ n r + ξ ′ (Ω − Ω s ) k Ω − c k ] k φ k dr ≥ Z r r [( µ + 1 r ) + 1 r + ξ ′ (Ω − Ω s ) ] k φ k dr > . (1-56)This contradicts Eq.(1-52). So proposition 1 is proved.Then, we prove proposition 2: if − ( µ + 1 /r ) < f ( r ) < c r = Ω s , there must be c i = 0 .Proof: If c i = 0 , then multiplying Eq.(1-53) by ( c r − c t ) /c i , where the arbitrary real number c t does not depend on r , and adding the result to Eq.(1-52), it satisfies Z r r r {k φ ′ k + [ n r + ξ ′ (Ω − c t ) r k Ω − c k ] k φ k } dr = 0 . (1-57)But the above Eq.(1-57) can not be hold for some special c t . For example, let c t = 2 c r − Ω s , thenthere is (Ω − Ω s )(Ω − c t ) < k Ω − c k , and ξ ′ (Ω − c t ) k Ω − c k = f ( r ) (Ω − Ω s )(Ω − c t ) k Ω − c k > − ( µ + 1 r ) . (1-58)This yields Z r r [ r k φ ′ k + ( n r + ξ ′ (Ω − c t ) k Ω − c k ) k φ k ] dr > , (1-59)which also contradicts Eq.(1-57). So the second proposition is also proved.Using ’proposition 1: if − ( µ + 1 /r ) < f ( r ) < c r = Ω s ’ and ’proposition 2: if − ( µ +1 /r ) < f ( r ) < c r = Ω s then c i = 0 ’, we find a stability criterion.Theorem 2: If the velocity profile satisfy − ( µ + 1 /r ) < f ( r ) < U in a frame rotating with angular velocity Ω . Acriterion is found that instability occurs locally when 2Ω(2Ω − U ′ ) < U ′ = dU/dr representsradial shear of horizontal velocity. Pedley’s criterion, which is recovered by later researches [38, 39],is in essence the special case of Rayleigh’s circulation criterion, i.e., ddr (Ω r ) < hapter 1 General Stability Criteria Here the proof is briefly given. Considering the narrow-gap approximation r − r = d ≪ r andthe large radii approximation 1 /r → ′ r ≈ − U ′ . SoΦ = d (Ω r ) /dr/r = 2Ω(2Ω − U ′ ) < r − r = d ≪ r and the large radii approximation 1 /r → § As well known, these two kinds of instabilities are independent with each other. Howard [6, 36, 41]has given an example which is stable to axisymmetric disturbances but unstable to two-dimensional(shear) disturbances. Here another example is given that the flow is unstable to axisymmetric distur-bances but stable to two-dimensional disturbances according the stability criteria above. To illuminatethis, a simple example is given as Ω( r ) = 1 − r with the vorticity ξ = 2 − r . As shown in Fig. 1-6,both Ω (solid line) and ξ (dashed line) are plotted within the interval r ≤ r ≤ r . It is fromRayleigh’s inflection point criterion that the flow is always stable to two dimensional disturbances.While the flow is unstable for the axisymmetric disturbances, if 2 / < r < / < r < N > U ′ / N is known as Brunt-V¨ais¨al¨a frequency, U is horizontal velocity r Ω , ξ Ωξ Figure 1-6: Angular velocity Ω (solid line) and corresponding vorticity ξ (dashed line) versus r .20 and U ′ = dU/dz represents vertical shear of horizontal velocity. This criterion is then an analogueof the following criterion by Howard and Gupta [36]: The rotating flow is stable for axisymmetricdisturbances if Eq.(1-60) holds,Φ ≥
14 ( dWdr ) , (1-60)where W is the axis velocity, and it vanishes in the pure rotating flows, as in Rayleigh’s criteria.Comparing the two stability criteria, the analogues of Φ and dW/dr are N and V ′ , respec-tively. Physically the rotating establishes a potential distribution along the ratio, which is denoted byRayleigh discriminant Φ . Similarly, the stratification establishes a potential distribution along thegravity, which is denoted by Brunt-V¨ais¨al¨a frequency. And the analogue of the radial shear in rotatingflow is right the vertical shear in stratified flow. So the criteria between rotation and stratificationcan also be analogy with each other in this way.In short, the general stability criterion is obtained for inviscid rotating flow. Then Pedley’scirterion is proved to be an special case of Rayleigh’s criterion. These results extend Rayleigh’sinflection point theorem for curved and rotating flows, and they are analogues of the theorems provedby Fjørtoft and Sun for the two-dimensional inviscid parallel flows. § Barotropic waves are widely existed in the flow with shear. However, the general theory forbarotropic waves still lacks though the stability of linear waves (e.g. Rossby wave) are always themost concerned. The theory of wave-mean flow interaction points out that the flow is unstable, thedisturbed waves can grow by taking energy from the mean flow. However, why the disturbance wavescan’t take energy from the mean flow in the stable flow is not known. Nevertheless, the general criterioncan be great helpful to understand these problems. As Arnol’d’s criteria deal with the total energy,they can not provide such kind of information. In this way, a simply linear criterion correspondingto Arnol’d’s second nonlinear stability theorem is necessary for considering the stability of waves. Italso provides a new way to investigate shear instability. The aim of this short paper is to find such astability criterion from Rayleigh’s equation in normal mode way, through which the shear instabilitymay be understood.The above criteria would be helpful for understanding the wave-mean flow interaction, especiallythe Rossby wave-mean flow interaction in barotropic flows. According to the stable criteria, thenecessary condition for wave-mean flow interaction can be obtained. And why the disturbed wavescan’t take energy from the mean flow in the stable flow is revealed. If the flow is stable, there is nowave-mean flow interaction at all.First, when the velocity profile has no inflection point, then the speeds of barotropic waves c r − U = 0 . According to Eq.(1-3), U − c r > U ′′ < U if the vortex gradient is positive. This extendsthe west-propagation theory of Rossby waves. And U − c r < U ′′ > U if the vortex gradient is negative.Second, when the velocity profile has an inflection point U ′′ s = 0 , the speed of most favorite wavewhich might have interaction with the mean flow should be c r = U s . However, the wave speed c r = U s holds for − µ < U ′′ U − U s < hapter 1 General Stability Criteria can be easily generated to the flows in β plane. Thus, the waves have no interaction with the stableflows. This is the reason why the disturbed waves can’t take energy from the stably mean flow.On the other hand, [15] pointed out that c r of a unstable wave must lie between the minimumand the maximum values of the mean velocity profile. Thus, the unstable wave is stationary relativeto the mean flow, and can take energy from the mean flow. So there are wave-mean flow interactionsin unstable flows.In summary, the general stability criterion is obtained for inviscid both parallel and rotatingflows. Then Pedley’s cirterion is proved to be an special case of Rayleigh’s criterion. These resultsextend Rayleigh’s inflection point theorem for curved and rotating flows, and they are analogues of thetheorems proved by Rayleigh, Tollmien and Fjørtoft for the two-dimensional inviscid parallel flows.Besides, this would intrigue future research on the mechanism of hydrodynamic instability.22 hapter 2 2 Horizontal Convection § § The abyssal ocean circulation is thought of an important energy conveyor belt, which has greatimpact to climate change [42, 43, 44]. As former investigators noted the fact that density watersinks by surface cooling at North Atlantic Ocean, this is so called density-driven flow. Horizontalconvection, in which the flow is uneven heated at the horizontal surface, was taken as a model of suchcirculation. However, a novel idea emerges when the ocean energy balance is considered [45, 46, 47].The horizontal convection become a key model to exam the theories. Two classes of flows are oftenused. The first one, named as two-basin forcing flows hereafter, has symmetric surface forcing likethat of South and North Atlantic Ocean basins. The multiple equilibria and bifurcation phenomenaof such flows are often discussed, especially in numerical way [48, 49, 50, 51]. The second one, namedas one-basin forcing flows hereafter, has monotone forcing from one side to another like that of NorthAtlantic Ocean basin. This kind of flow is more conveniently used in the experiments [52, 53, 54].Some general properties of horizontal convection have been obtained by theoretic studies. Aboveall, [52] found the 1/5-power laws of Rayleigh number Ra for flow strength Ψ max and the Nusseltnumber N u , which are consistent with experiments and numerical simulations. This 1/5-power lawof Ra is generally valid no mater what the flow is steady or non-steady [e.g. 48, 52, 53, 54, 55, 56].Thus it is useful to consider what the horizontal convection would be as Ra → ∞ . [51] focused on theenergy dissipation under the conditions of the viscosity ν and thermal diffusivity κ being loweredto zero with Prandtl number Pr fixed. They proved that the horizontal convection is non-turbulentunder certain definition of turbulence. Motivated by above results, [56] claimed N u is bounded by Ra / as Ra → ∞ .On the other hand, numerical simulations and experiments were also used to find details of theflows. [52, 55] found by both experiment and numerical simulations that the flows are steady andstable in one-basin circulation. Comparing to Rossby’s one-basin forcing, [48] studied the multipleequilibria under symmetric two-basin forcing numerically. They also claimed that the flow is alwayssymmetric if only theromal or salt forcing at least for their parameter regimes. If both kinds of forcingare considered, there will be symmetry breaking and even unsteady flows [e.g. 48, 49, 50, 57]. Forone-basin circulation, it was thought that horizontal convection must be steady and stable even for23 hapter 2 Horizontal Convection Ra → ∞ , according to Sandst¨orm’s theorem [e.g. 46, 58, 59].But this scenario seems to be violated in recent investigations. [51] studied the horizontal convec-tion under the symmetric two-basin forcing. The smaller the Prandtl number is, the more unstablethe flow is. As the Prandtl number increases to 10, the circulation tends to be a shallow cell. Theyalso obtained the thresholds for the transition from steady flows to unstable and steady flows. Theseunsteady flows then also ware reported in experiments [53]. However, such unsteady convective mix-ing and turbulent interior motion in the exmpriments are nonturbulent according to the definition ofturbulence [51]. As the plume rises from bottom to top and through the full depth of the tank, thiskind of flows are referred as full-penetrating flows. While [54] showed in their experiments a totallydifferent result. The motion of the circulation, though being visible to the naked eye, is vanishinglysmall. The convection cells, shallow and near the heating surface, are steady and stable. This kind offlows are quite different from those in [53], and they were referred as partial-penetrating flows [54].Though lots of numerical simulations on the horizontal convection, none of them have obtainedsuch partial-penetrating flow and the onset of such flow needs to reveal yet. The main purpose of thispaper is to investigate the partial-penetrating flow by numerical simulation, thus resulting in a morecomprehensive view on this issue. The model and the scheme are sketched in § § § § § We consider the the horizontal convection flows within the two-dimensional domain, and Boussi-nesq approximation is assumed to these flows. The horizontal (y) and vertical (z) regimes are0 ≤ y ≤ L and 0 ≤ z ≤ D , respectively. Similar to [51], the depth D is taken as reference lengthscale and A = D/L denotes the aspect ratio. We use A = 1 in present work, which is consistentwith the experiments by [54]. Taking account of nondivergence of velocity filed in Boussinesq approx-imation, the lagrangian streamfunction Ψ and the corresponding vorticity ω are introduced. Thevelocity −→ u = ( v, w ) , where horizontal velocity v = ∂ Ψ ∂z and vertical velocity w = − ∂ Ψ ∂y , respectively.The governing equations [48, 49, 50, 51, 56] in vorticity-streamfunction formulation are ∂T∂t + J (Ψ , T ) = ( ∂ T∂y + ∂ T∂z ) (2-1a) ∂ω∂t + J (Ψ , ω ) = − Pr( ∇ ω + Ra ∂T∂y ) (2-1b) ∇ Ψ = − ω (2-1c)where J (Ψ , φ ) = ∂ Ψ ∂y ∂φ∂z − ∂φ∂y ∂ Ψ ∂z denotes the nonlinear advection term. There are two important di-mensionless parameter in Eq.(2-1), i.e. Rayleigh number Ra = α T ∆ T gL / ( κν ) and Prandtl numberPr = ν/κ , where g , α T , ∆ T , L , κ and ν are gravity acceleration, thermal expansion coefficient,surface temperature difference, length of horizontal domain, thermal diffusivity and kinematic viscos-ity, respectively. The surface buoyancy forcing is T = sin( π y ) , so that only one-basin flows insteadof symmetric two-basin flows can be obtained.There are two important quantity describing the circulation, i.e. the non-dimensional stream-function maximum and the non-dimensional heat flux. The non-dimensional streamfunction maxi-24 mum Ψ max = Ψ ∗ max /ν , where Ψ ∗ max is the maximum of the dimensional streamfunction. For thenon-dimensional heat flux is defined as f T = ∂T /∂z at the heated surface.The above Eq.(2-1) is solved with Arakawa scheme [e.g. 60, 61] and non-uniform grids. Comparingto the other schemes, Arakawa scheme is more accuracy but more expensive, and it has also beenapplied to horizontal convection flows at high Rayleigh number [62].To validate the scheme, we calculate the nature convection problem with the resolution of 80 × §
1. the establishment of circulation
Similar to the experiments by [54], we use A = 1 for the numeric simulations, the circulationsare obtained for Ra > . The dimensionless time t = t ∗ /τ , where t ∗ and τ = D /κ are thedimensional and the unit scaling times, respectively. A typical value of t = 1 is about 80 hours in thedimensional time, given D = 20 cm and κ = 1 . × − cm /s , which are approximate to the valuesused in the experiments by [53, 54]. According to the flow pattern, the establishment of the circulationcan be divided into three stages: (1) startup of circulation, (2) damp of secondary circulation, (3)amplification of primary circulation. The former two stages are relatively short, while the third oneis very long. At the end of last stage, the circulation is fully established.To illuminate this, the circulation of Pr = 8 and Ra = 5 × is taken as an example. Duringthe first stage, the circulation is established as soon as the surface forcing is superposed (Fig.2-1 a ).The gradient of horizontal buoyancy drives the water like a lid, so that a very energetic primarycirculation ( Ψ max >
56 ) generates (Fig.2-1 b ), which was also observed by the experimentalist [54].But this process is much more faster, and soon the secondary circulation emerges below the primarycirculation at t = 3 . × − (Fig.2-1 c ). The secondary circulation becomes stronger and stronger,and the primary circulation becomes weaker and weaker. At the end of the stage, the primaryauthor Ra Ψ mid Ψ max u max ( y ) v max ( x ) N u
Present 10 .
430 16 .
863 64.47(0.85) 219.17 (0.038) 8 . .
386 16 .
811 64.83(0.85) 220.56 (0.038) 8 . .
386 16 .
811 64.83(0.85) 220.57 (0.038) 8 . .
586 30 .
426 146.05(0.886) 687.17 (0.022) 16 . .
361 30 .
165 148.59(0.879) 699.17 (0.021) 16 . .
356 30 .
155 148.57(0.879) 699.17 (0.021) 16 . .
71 . Ψ mid , Ψ max are the values in the midpoint and the maximum of streamfunction, respectively. u max and v max arethe maximum of horizontal and vertical velocity component on the vertical midplane x = 0 . y = 0 . y and x , respectively. And N u is average Nusselt number at theheated wall. The resolution is 80 ×
80 meshes for present results.25 hapter 2 Horizontal Convection y z ( a ) t =1.6*10 -4 - - y z ( d ) t =4.8*10 -4 - -10 0 y z ( e ) t =6.4*10 -4 - -15 0 y z ( f ) t =8*10 -4 y z ( b ) t =2.4*10 -4 y z ( c ) t =3.2*10 -4 Figure 2-1: Snapshots of the flow fields (streamfunction Ψ ), with solid counter curves for Ψ > < y z ( a ) t =1.6*10 -4 . y z ( d ) t =4.8*10 -4 y z ( e ) t =6.4*10 -4 y z ( b ) t =2.4*10 -4 y z ( f ) t =8*10 -4 y z ( c ) t =3.2*10 -4 Figure 2-2: Snapshots of the temperature fields corresponding to these in Fig.2-1, with counter inter-vals being 0.1. 26 circulation becomes partial-penetrating, the secondary circulation becomes full-penetrating (Fig.2-1 f ). Meanwhile, the heat conducts from top to bottom along the side walls (Fig.2-2).Comparing to the first stage, the second one is longer, during which the secondary circulationdamps and breaks into several weaker circulations (Fig.2-3). In consequence, the smaller circula-tions emerges from the bottom one after another. And the flow field fulfills such partial-penetratingcirculations, as observed by [54]. The damp of the secondary circulation is a puzzle for the experi-mentalists, as the corresponding temperature field was not well measured then. Here the temperaturefields (Fig.2-4) explains the reason why such full-penetrating circulation can’t be maintained. Thelack of driven forcing and the viscous friction are the reasons. As in Fig.2-4, there is vanishingly smallbuoyancy gradient in the secondary circulation, so that it damps down due to viscosity and boundaryfriction, especially the circulation near the bottom. Meanwhile, the primary circulation covers theconvection region, where the horizontal buoyancy gradient is remarkably large.The last stage is a long and slowly process of approaching to quasi-equilibrium state. The primarycirculation is amplified in this stage, and the secondary circulations disappear from the bottom, liketheir emerging process (Fig.2-6). It is notable that the primary circulation (e.g. the shadowed shallowcirculation cell in Fig.2-6) seldom changes during this stage, which implies that the flow near surfaceapproaches to quasi-equilibrium state relatively faster. While the process to quasi-equilibrium stateis very slowly near the bottom. Comparing the flow field and temperature field, it is clear that theflow field can’t be steady until thermal conduction is balanced.The above stages have different time scales. The startup stage is the most fast stage, duringwith the circulations are established within t = 10 − (several minutes in laboratorial time). The
00 016 y z ( d ) t =9.6*10 -4
00 -1 y z ( e ) t =1.28*10 -2 -2 -1 y z ( c ) t =6.4*10 -3
00 -1 y z ( f ) t =1.6*10 -2
16 -2-3 y z ( b ) t =3.2*10 -3 - -
16 08 y z ( a ) t =1.6*10 -3 Figure 2-3: Snapshots of the flow fields (streamfunction Ψ ), with solid counter curves for Ψ > < > < hapter 2 Horizontal Convection . y z ( d ) t =9.6*10 -3 y z ( e ) t =1.28*10 -2 y z ( f ) t =1.6*10 -2 y z ( a ) t =1.6*10 -3 y z ( b ) t =3.2*10 -3 y z ( c ) t =6.4*10 -3 Figure 2-4: Snapshots of the temperature fields corresponding to these in Fig.2-3, with counter inter-vals being 0.1.second stage and the last state are time scales of t = 10 − (several hours) and t = 2 . E k and Ψ max as indexes to overseethe evolution of the circulations, where E k = H ( v + w ) / dy dz . It is notable that the time ap-proaching to quasi-equilibrium state is very long (about one week) in our numerical simulations, butis is remarkably shorter in the experiments [53, 54]. According to the parameters used by [54], not allof the experiments take enough time, so that the circulations are not fully established in some cases.This is more serious in the experiments by [53], where the experiment time of 30 hours is much lessthan t e of 200 hours.The establishment time t e of the circulation is defined as the time when d Ψ /dt < . max holds for whole field. It is found that the partial-penetrating circulation is established very fast, butit takes a very long time for the flow to approach to equilibrium state. Fig.2-7 displays the timeevolutions of total kinetic energy E k and Ψ max with time t . The circulation is established very fast,and both E k and Ψ max reach their 95% of the equilibrium values within t = 0 .
05 (4 hours). Inaddition, Fig.2-7 b agrees well with the experiments (e.g. Fig.2 in [54]), and this fast establishment ofthe circulation was also noted by [53]. However the total time to get the quasi-equilibrium state isrelatively longer, it takes at least t e = 2 . t e = 2 . ≤ Pr ≤
10 and 10 ≤ Ra ≤ . It also implies that the establishment ofthe circulation is governed by conduction process along the side walls, as t e is only determined by D and κ . The conclusion of t e = 2 . y z ( a ) t =0.5 y z ( b ) t =0.6
20 0 y z ( c ) t =0.7
10 020 y z ( d ) t =1.0 y z ( e ) t =1.5 y z ( f ) t =2.5 Figure 2-5: Snapshots of the flow fields (streamfunction Ψ ) for Pr = 8 and Ra = 5 × , withsolid counter curves for Ψ > < > y z ( a ) t =0.5 y z ( c ) t =0.7 y z ( e ) t =1.5 y z ( f ) t =2.0 . y z ( d ) t =1.0 y z ( b ) t =0.6 Figure 2-6: Snapshots of the temperature fields corresponding to these in Fig.2-5.29 hapter 2 Horizontal Convection t ( b ) Ψ max t E k ( a ) Figure 2-7: The time evolution of the flow field for total kinetic energy E k ( a ) and maximum flowΨ max ( b ) at P r = 8 and Ra = 5 × .The fast established but slowly steadied flow is due to the horizontal evolution of buoyancy (ortemperature). It is well known that the horizontal gradient of buoyancy drives the circulation, whichcan also be known from Eq.(2-1 b ). Thus the fast establishment of the temperature gradient near thesurface makes establishment of the circulation very fast. To illuminate this, the temperature fieldsof the circulation are shown in Fig.2-6. The horizontal temperature gradient is relatively large nearthe top surface, but it is relatively to small to be vanished near the bottom. Comparing Fig.2-5 a with Fig.2-6 a , the main circulation is right within the zone where temperature gradient is remarkablylarge, so as to the figures at other times. As the time goes on, the main circulation becomes deeperand deeper with the downward propagation of buoyancy gradient. So the slowly heat conduction fromthe top surface to the bottom makes the establishment of the circulation near bottom to be a longtime process. Hence, the depth D of the tank and the thermal diffusivity κ both determinate theestablishment time t e of the circulation. And it is from Fig.2-7 that t e is 2.5 for the circulation ofPr = 8 and Ra = 5 × .
2. power laws
As the parameters in different ocean global circulation models (OGCM) varies in wide regime[e.g. 47, 65], it is very useful to know how the sensitivity of the results to the parameters. This is alsouseful when the experimental results are extrapolated to real ocean circulations [53].Then, the power laws at different Prandtl numbers (e.g. Pr = 1 , Pr = 4 , Pr = 8 and Pr = 10 )are calculated to investigate the sensitivities of
N u and Ψ max to Pr . Fig.2-8 a and Fig.2-8 b showthe 1/5-power laws of Ψ max and N u , respectively. Noting that Ψ max instead of Ψ max and Ra p = RaP r / instead of Ra are in Fig.2-8 a , it implies that the flow is dominated by thermal diffusivity κ . The larger Prandtl number is, the stronger the flow is. All the direct numeric simulation (DNS)data at Pr = 1 , Pr = 4 , Pr = 8 and Pr = 10 lie in the line of this power law, Ψ max = 0 . Ra / p ,where Ra p = Ra Pr / . On the other hand, [54] fitted their experiments data into two different 1/5-power laws. It is amazing that more than half of the experimental data (14/25) by [54], of whichthe reduced acceleration due to gravity g ′ > . cm/s , lie in this line (right triangles in Fig.2-8 a ).Taking account that the aspect ratio in DNS is not exact the same as in their experiments, these dataare consistent well with each others. Meanwhile, the Nusselt number N u in DNS lie in two slightlydisplaced parallel lines, one for Pr = 1 and the other for Pr = 4 , Pr = 8 and Pr = 10 . It seemsthat Ψ max is more sensitive to Prandtl number than Nusselt number does.30 Ra p Ra p Ψ max ( a ) Ra Ra Ra Nu ( b ) Figure 2-8: The flow streamfunction Ψ max ( a ) and the heat flux N u ( b ) vs. Rayleigh number, where Ra p = RaP r / in ( a ).It is notable that 1/5-power law in Fig.2-8 b is something different with that of [56], where N u is very sensitive to Pr ( 0 . < Pr < N u is little sensitive to Pras Pr > b is consistent with the results of [55]. Here a simple explanation is presented forthese results.Considering the steady state solution of Eq.(2-1) as Ra → ∞ , we obtain the following equationby taking Ψ = Ψ ∗ /κ [48, 52]. J ( T, Ψ) = ( ∂ T∂y + ∂ T∂z ) (2-2a) P r − J ( ∇ Ψ , Ψ) = ( ∂ Ψ ∂y + ∂ Ψ ∂z ) + Ra ∂T∂y (2-2b)When Pr ≫ P r − J ( ∇ Ψ , Ψ) in Eq.(2-2 b ) can be ignored, and thebuoyancy forcing term Ra ∂T∂y balances the viscous term ∇ Ψ . In this case, the governing equationsis independent of Prandtl number Pr , which is due to [52]. This is the reason why
N u and Ψ max arelittle sensitive to Pr for Pr >
10 in the numerical simulations by [55]. When Pr is order of 1 or evenless than 1, the convection term can not be ignored and Pr plays a role in this case, as obtained by[56]. As Pr ≪ Ra numbers,which can be seen from Eq.(2-1 b ). This is also can explain the numerical results by [51], where theflow becomes unsteady and has strong eddy diffusion at Pr = 0 . § § Horizontal convection, in which the water is unevenly heated at the horizontal surface, was takenas a model of abyssal ocean circulation. The circulation, driven by density gradient and referred asthermohaline circulation (THC) at North Atlantic Ocean, is thought of an important energy con-31 hapter 2 Horizontal Convection veyor belt and has great impact to climate change [42, 43, 44]. The horizontal convection become animportant model to discuss the ocean energy balance [45, 46, 47]. Unlike the Rayleigh-B´enard con-vection, the horizontal convection can be set to motion by any small temperature gradient. Moreover,the horizontal convection yields 1/5-power laws of Ra , comparing with the 1/4-power laws in theRayleigh-B´enard convection.The 1/5-power laws of Ra for flow strength (streamfunction maximum Ψ max ) and the heat flux(Nusselt number N u ), first found by [52], were later approved by both experiments [e.g. 52, 53, 54]and numerical simulations [e.g. 48, 55, 56, 66]. According to the scaling analysis [48, 52], there isa boundary layer near the surface, which is inverse proportion to 1/5 power of Ra . Both the flowstrength and the heat flux are dominated by the scale of boundary-layer. Although this 1/5-powerlaw of Ra is obtained for steady flow, it is still valid even for unsteady flow [e.g. 53].The unsteady flow in horizontal convection was first found by numerically [51], then was observedin the experiment [53]. This unsteady flow is proved to be non-turbulent for that the energy dissipationturns to zero under the conditions of the viscosity ν and thermal diffusivity κ being lowered to zerowith Prandtl number Pr = ν/κ being fixed [51]. Motivated by above investigations, [56] tried to findthe bounded of N u as Ra → ∞ . However, their conclusion of N u ≤ cRa / (for some constant c )seems not tight enough as all the numerical simulations yield 1/5-power law.In a recent experiment, a new flow configuration referred as ”parti-penetrating flow” was reported[54]. According to the measurement, the circulation cell is shallow and and no longer occupies thewhole length of the tank. Though lots of numerical simulations on the horizontal convection, noneof them have obtained such partial-penetrating flow. The main reason is that all of the formersimulations used free slip condition on the walls [48, 55, 56, 62], the energetic circulation turns to befull-penetrating. While the laboratory experiments always require no slip on the walls [54], the viscousdrag slows down the vigorous circulation. Noting that the power law fitted from the experimental datais somehow coarse, this can be improved by numerical simulations. Moreover, the partial-penetratingflow and the onset of such flow needs to reveal too.The main purpose of this paper is to investigate the partial-penetrating flow and to find a moreaccurate power law by numerical simulation, thus resulting in a more comprehensive view on this issue.The model and the scheme are sketched in § § § § § § § We consider the the horizontal convection flows within the two-dimensional domain, and theBoussinesq approximation is assumed to these flows. The horizontal (y) and vertical (z) regimes are0 ≤ y ≤ L and 0 ≤ z ≤ D , respectively. Similar to [52], the depth L is taken as reference lengthscale and A = D/L denotes the aspect ratio. We use A = 1 in present work as [52] did, while A = 0 .
675 is used in the experiments by [54]. Taking account of nondivergence of velocity fieldin Boussinesq approximation, the lagrangian streamfunction Ψ and the corresponding vorticity ω are introduced. The velocity −→ u = ( v, w ) , where horizontal velocity v = ∂ Ψ ∂z and vertical velocity w = − ∂ Ψ ∂y , respectively. The governing equations [48, 49, 50, 51, 56] in vorticity-streamfunctionformulation are 32 ∂T∂t + J (Ψ , T ) = ( ∂ T∂y + ∂ T∂z ) (2-3a) ∂ω∂t + J (Ψ , ω ) = − Pr( ∇ ω + Ra ∂T∂y ) (2-3b) ∇ Ψ = − ω (2-3c)where J (Ψ , φ ) = ∂ Ψ ∂y ∂φ∂z − ∂φ∂y ∂ Ψ ∂z denotes the nonlinear advection term. There are two important di-mensionless parameter in Eq.(2-3), i.e. Rayleigh number Ra = α T ∆ T gL / ( κν ) and Prandtl numberPr = ν/κ , where g , α T , ∆ T , L , κ and ν are gravity acceleration, thermal expansion coefficient,surface temperature difference, length of horizontal domain, thermal diffusivity and kinematic vis-cosity, respectively. The boundary condition is the same with the experiment: the surface buoyancyforcing is T = sin( π y ) , and no slip boundary condition is applied to walls except for surface.There are two important quantity describing the circulation, i.e. the non-dimensional stream-function maximum and the non-dimensional heat flux. The non-dimensional streamfunction maxi-mum Ψ max = Ψ ∗ max /ν , where Ψ ∗ max is the maximum of the dimensional streamfunction. The non-dimensional heat flux is defined as f T = ∂T /∂z on the heated surface. Nusselt number N u , whichis defined here as the maximum of ∂T /∂z on the top surface. This definition of
N u is somethingdifferent from the others [e.g. 53, 56].The above Eq.(2-3) is solved with finite different method in non-uniform grids. Crank-Nicholsonscheme and Arakawa scheme [e.g. 60, 61] are applied to discretize the linear and nonlinear terms,respectively. Comparing to the other schemes, Arakawa scheme is more accuracy but more expensive,and it has also been applied to horizontal convection flows at high Rayleigh number [62].To validate the scheme, we calculate the nature convection problem with the resolution of 80 × §
1. spatial resolution
First, we investigate grid dependency of the solutions to ensure that the numerical simulationsare valid. The boundary condition is the same with the experiment: the surface buoyancy forcing is T = sin( π y ) , and no slip boundary condition is applied to walls except for surface. To this purpose,a case of Ra = 2 × and Pr = 1 is calculated with grids of three different resolution, i.e. thehorizontal number of meshes N = 40 , N = 64 and N = 80 . We find that the resolution of gridsmust be fine enough, otherwise some unphysical time-depend solutions would be obtained.Fig.2-9 a depicts the time evolution of the maximum Ψ max . The solutions tend to be steady astime t > N = 64 and N = 80 . While it becomes time-dependent for N = 40 . It impliesthat some unphysical time-dependent solutions might be obtained if the spatial resolution is not fineenough. To exclude the unphysical time-dependent solutions, the numerical simulations must beobtained with sufficient spatial resolution which depends on the Rayleigh number Ra . As Fig.2-9 b shows, the minimal number of horizontal meshes N is to obtain correct results directly proportion toRa / . Taking account of Ra ∝ L , this means N ∝ L : the longer L is, the larger N is. To obtain33 hapter 2 Horizontal Convection the physical solutions, N must be within the stable regime in Fig.2-9 b . According to our calculations,the flow is still steady and stable for Ra ≤ .It is from Fig.2-9 b that ∆ y = L/N = C R ( κν ) / / ( α T ∆ T g ) / , where ∆ y and C R = 10 are themesh size in y direction and the coefficient, respectively. The smaller κ and ν are, the smaller themesh should be. For the molecular kinematic viscosity ν = 1 . × − cm /s and thermal diffusivity κ = 1 . × − cm /s in the case of “run 16” by [54], the mesh ∆ y should be 2 . mm , which issmaller than Kolmogorov scale η = ( ν /ǫ ) / = 5 . mm , where ǫ = 2 × − cm /s is dissipation ratein the field [54]. So this implies that the mesh should be fine enough to resolute Kolmogorov scaleeddies.The resolution requirement of ∆ y implies that the bound of L in numerical simulations is aboutlaboratory scale if molecular viscosity and diffusivity are used. And eddy viscosity and diffusivity arerequired, when the length of ocean scale is considered.
2. the establishment of circulation
Then the circulations are obtained numerically for Ra > . Similar to [51], the dimensionlesstime t = t ∗ /τ , where t ∗ and τ = D /κ are the dimensional and the unit scaling times, respectively.A typical value of t = 1 is about 80 hours in the dimensional time, given D = 20 cm and κ =1 . × − cm /s , which are approximate to the values used in the experiments by [53, 54]. In thefollowing simulations, we use the total kinetic energy E k and Ψ max as indexes to oversee the evolutionof the circulations, where E k = H ( v + w ) / dy dz . It is found that the flow is established very fast,but it takes a very long time for the flow to approach to equilibrium state.To illuminate this, the circulation of Pr = 8 and Ra = 5 × is taken as an example. Fig.2-7 displays the time evolutions of total kinetic energy E k and Ψ max with time t . The circulationis established very fast, and both E k and Ψ max reach their 95% of the equilibrium values within t = 0 .
05 (4 hours). In addition, Fig.2-7 b agrees well with the experiments (e.g. Fig.2 in [54]), andthis fast establishment of the circulation was also noted by [53]. However the total time to get thequasi-equilibrium state is relatively longer, it takes at least t e = 2 . t N=40N=64N=80 ( a ) Ψ max N
50 100 150 20010 Ra N stableunstable ( b ) Figure 2-9: ( a ) The maximum of streamfunction Ψ max vs time t for Ra = 2 × . The solid, dashedand dash-doted curves are solutions with N = 40 , N = 64 and N = 80 , respectively. ( b ) The stableand unstable regime on the plot of Rayleigh number Ra vs N .34 -0.3752010 - . y z ( b ) -0.2250 y z ( c ) -0.005 - . -0.0002 y z ( a ) y z ( d ) Figure 2-10: The flow fields (streamfunction Ψ ) at four different time steps: t = 0 . a ), t = 1 . b ), t = 1 . c ) and t = 2 . d ) at Pr=8 and Ra = 5 × , solid curves for clockwise flow ( Ψ > < >
20 )are shadowed and the counter intervals are 10 for Ψ > hapter 2 Horizontal Convection approaches to equilibrium state within t = 1 , as Fig.2-10 a and Fig.2-10 b show. Meanwhile, there areseveral very weakly clockwise and anticlockwise circulation cells below the primary circulation cell,which are secondary flows due to the drag by upper primary circulation. These secondary cells werealso observed in the experiments by [54]. As the time goes on, the primary circulation cell becomesstronger and larger, and the secondary circulations are weaker and smaller. So that the counter lineof Ψ = 10 becomes deeper and deeper as shown from Fig.2-10 a to Fig.2-10 d . Finally, the primarycell fulfills the whole tank at t = 2 . d ).Then, the flows within 10 ≤ Ra ≤ are calculated. Fig.2-8 a shows the 1/5-power law of Ra for Ψ max at different Prandtl numbers (e.g. Pr = 1 , Pr = 4 , Pr = 8 and Pr = 10 ). Notingthat Ra p = RaP r / instead of Ra are used in Fig.2-8 a , it implies that the flow is dominated bythermal diffusivity κ . The larger Prandtl number is, the stronger the flow is. All the direct numericsimulation (DNS) data at Pr = 1 , Pr = 4 , Pr = 8 and Pr = 10 yield,Ψ max = 0 . Ra / p = 0 . Ra / P r / (2-4)On the other hand, [54] fitted their experiments data into two different 1/5-power laws: bigger onefor the reduced acceleration due to gravity g ′ > . cm/s , the smaller one for g ′ < . cm/s .Equation (2-4) is similar to but more accurate than the bigger one obtained by [54]. It is notablethat all the experimental data lie around this line (right triangles in Fig.2-8 a ), as the Prandtl numberis considered. Taking account that the aspect ratio in DNS is not exact the same as that in theirexperiments, these data are consistent well with each others. Fig.2-8 b shows the 1/5-power law for N u . This is consistent with the scaling analysis, i.e. the thermal boundary-layer is inverse proportionto Ra / . It is obvious that N u is less sensitive to
P r than that of Ψ max , so that
N u seldomchanges for Pr >
3. the partial-penetrating cell
It is notable that the main circulation near the surface is seldom changed during the establishmentprocess, especially the close circulation cell shadowed in Fig.2-10. As this cell is shallow and thecirculation is only near the surface, it is referred as the ”partial-penetrating cell” after [54]. Forexample, the shadowed cell height is about 1/6 of the total depth in Fig.2-10, so that the whole cellis within the boundary-layer near the surface. This kind of flow also exist in other Rayleigh numbers,where the shallow and close cells like that in Fig.2-10.An objective definition of ”partial-penetrating cell” is more convenient for further discussion. Inpresent investigation, the penetrating depth of the cell D c is defined as the depth of a close circulationcell, which contains 60% of the total amount. Here the concept of partial-penetrating cell has suchmeanings: (1) above all, the close circulation cell is very shallow comparing to its width, so that D c is within the boundary-layer near the surface, (2) consequently, the close cell is seldom affected by thebottom boundary, (3) moreover, the flow in close circulation cell is dominant of the main circulation,e.g., the flow rate in the shallow cell is about 60% of the total amount (e.g. Ψ c = 20 in Fig.2-10).On the contrary, ”the full-penetrating circulation” is referred to the circulation takes whole depth ofthe tank hereinafter.Using the above definition, the shadowed cell in Fig.2-10 is the partial-penetrating cell. First,it is obvious that the shadowed cell satisfies conditions (1) and (3). Second, the close cell is seldomaffected by the bottom boundary. To illuminate this, we descend the depth of the water tank (orequally descending aspect radio A ). Fig.2-11 clear depicts that the partial-penetrating part is little36 y z ( c ) y z ( b ) y z ( a ) Figure 2-11: The flow fields (streamfunction Ψ ) near the forcing surface of three respective aspectratios: A = 1 ( a ), A = 0 . b ) and A = 0 .
25 ( c ) at Pr=8 and Ra = 5 × . The partial-penetratingcells are shadowed and the counter intervals are 10 in each figures.sensitive to bottom, if the depth of the tank is large enough ( D c < D < D c in Fig.2-11 c ). It mayalso be noted that the flow in Fig.2-11 c is very similar to the flow field near the surface ( 0 . ≤ z ≤ a . For that at that time in Fig.2-10 a , the boundary of main circulation Ψ = 0 is about z = 0 .
75 , hence the flow near the top is approximation of the case in Fig.2-11 c . This also implies thatthe shadowed cell in Fig.2-10 is partial-penetrating.As the partial-penetrating cell is near the forcing surface, it is quite independent of the flowsin the middle and the bottom. First, the partial-penetrating cell changes little even when the full-penetrating circulation emerges in Fig.2-10 c , d . Second, In addition, Fig.2-11 clear depicts thatthe partial-penetrating part is little sensitive to bottom, if the depth is large enough. Thus thepartial-penetrating cell ( Ψ > Ψ c ) is unchanged given the tank is deep enough. On the contrary, thefull-penetrating part is very sensitive to the bottom, as the flow this part of flow approaches to thebottom.
4. the constrain for partial-penetrating flow
However, not all the flows have partial-penetrating cells like that in Fig.2-10, e.g. the flow atPr=1 and Ra = 5 × in Fig.2-12. It is obvious that the circulation of Pr = 1 , which is mainlyfull-penetrating, is more deep comparing to Pr = 8 . So only the close circulation cell for Ψ >
35 isshallow enough, and it is about 1 / >
35 is not like the partial-penetrating cell.37 hapter 2 Horizontal Convection y z Figure 2-12: The flow fields (streamfunction Ψ ) for full-penetrating flow at Pr=1 and Ra = 5 × .The counter interval is 5, and Ψ >
35 is shadowed. y z ( c ) y z ( b ) y z ( a ) Figure 2-13: The flow fields (streamfunction Ψ ) near the forcing surface of three respective aspectratios: A = 1 ( a ), A = 0 . b ) and A = 0 .
25 ( c ) at Pr=1 and Ra = 5 × . The streamfunctionΨ >
35 are shadowed and the counter intervals are 5 in each figures.38 Ra Ra -0.15 Ra -0.2 Ra -0.18 D c Figure 2-14: D c vs. Ra. The solid, dashed and dash dotted lines are power laws of Ra respectivelyfor D c at Pr = 6 , Pr = 4 and Pr = 2 .Then a new problem emerges. As there might be full-penetrating flows, what’s the constrain forthe partial-penetrating flow? To investigate this, D c is employed as an index of partial-penetratingflow according to the above definition. If D c satisfies − / Ra , then the flow iswithin the boundary-layer and the flow is partial-penetrating, for that the boundary-layer satisfies − / Ra [48, 52, 53]. Else if D c does not satisfy − / Ra , then theflow is full-penetrating. Thus, according to the power law for D c , the flow can be referred as eitherpartial-penetrating or full-penetrating.In fact, the critical parameter governing the exitance of partial-penetrating cell is Prandtl numberPr . The larger the Prandtl number is, the more obvious the partial-penetrating cell is. It is fromFig.2-14 that the smaller Pr is, the bigger or deeper D c is. Only for the flows at Pr ≥ D c satisfy the − / Ra . D c seldom changes with Ra even for the flows at Pr = 1 .It is notable that partial-penetrating flows exist for Pr ≥ ≫ P r − J ( ∇ Ψ , Ψ) in Eq.(2-3 b ) canbe ignored, and the buoyancy forcing term Ra ∂T∂y balances the viscous term ∇ Ψ . In this case, thegoverning equations is independent of Pr and the boundary layer is inverse proportion to Ra / .When Pr is order of 1 or even less than 1, the convection term can not be ignored. So that the strongconvection is full-penetrating as depicted in Fig.2-12.In words, it is Pr that governs the flow pattern. When Pr ≥ ≤ § The experiments of horizontal convection in a rectangle cavity are simulated numerically. Thesimulations agree well with the experimental data, and a more extensive 1/5-power law of Ra isobtained by fitting the DNS data. The partial-penetrating flows are revisited by numerical simulations.It is Pr that governs the the existent of the partial-penetrating flow. When Pr ≥ ≤ hapter 2 Horizontal Convection § § The power laws in horizontal convection is one of the fundamental for understanding the flows.It is reported from [52] and [54] that the flow strength (the maximum of stream function Ψ max ) isalways increasing with Ra / . However, the flux of the flow can’t be measured very accuracy, andthe parameters are bounded within Ra < due to experimental inconvenientness. Moreover, thenumerical simulations of [51] ( 0 . < P r <
10 ), [56] ( 0 . < P r < Ra used in numerical simulation are lower than themaximum value used in the experiments. It is found that Ψ max ∼ Ra / to Ψ max ∼ Ra / , and N u ∝ Ra / [56], which is consist with the results by [62] at P r = 10 . The flow strength dependscontinuously to Ra from Ψ max ∝ Ra to Ψ max ∝ Ra / .On the other hand, Paparella and Young (2002) [51] have claimed that the flux trends to zeroas thermal diffusion trends to zero and P r is fixed. Besides, SIggers et. al (2004) obtained that themaximum of heat flux
N u is proper to Ra / , which is not supported by their numerical simulations.Then two possibilities are suggested. First, their theory is not accuracy. Second, the numericalsimulation is not accuracy.Motivated by the above problem, the power laws of horizontal convection is investigated. Similarto the previous investigations, we consider the horizontal convection flows within the two-dimensionaldomain with aspect ratio A = 0 . § When the Rayleigh number is relatively lower, the flow is weak and the heat transportation isdominated by heat conduction [48, 56]. As the Rayleigh number increases, the thermal convectionenhances to balance the surface heat flux. Fig.2-15 shows the stream function and temperature fieldof Ra = 10 . There is a weak upwelling flow in the center and two strong downwelling flow in thetwo sidewalls, which are driven by central heating and sidewall cooling. Accordingly, the temperaturefield deforms it horizontal isothermal lines as the convection. There is a warm tongue (shadowed in . . . - . - . - . (a) . . . . . (b) Figure 2-15: The flow stream function (a) and temperature field (b) of Ra = 10 . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positive and negativevalues, respectively. 40 Fig.2-15b), which is due to the vigorous central flow near the top surface and the central flow near thebottom bringing the cold water to the middle. For example, the isothermal line of T = 0 . Ra = 10 , which are something different from the flow of Ra = 10 .First, the downwelling flow departs into two separate parts: one descends the middle, the other candescend to the bottom. Thus there are two vortex centers in the main circulation as in Fig.2-16a.Besides, there is a reverse secondary circulation along the bottom near the sidewalls ( 1 /
10 departsfrom sidewall), which have no mass interchange with the main circulations (dead water). Consequently,they have no contribution to the main thermal convection. Similar to Fig.2-15a, the isothermal linesare concentrated and horizontal in the center. Moreover, the isothermal lines in Fig.2-16b showsthere are a temperature boundary layer and a thermocline near the surface. In total, the flow fieldare dominated by cold water, where
T < . § The above investigation have obtained the flow pattern of two different Rayleigh numbers. Thegeneral properties of the horizontal flows are addressed here, especially for the sensitivity of Ra. Tothis purpose, the power laws for flow strength and heat flux are calculated.First, the flow strength vs. Ra is investigated, which is the main parameter of intensity in thethermohaline circulation. It is from Fig.2-17a that the flow strength Ψ max is linear proper to Ra as Ra < , and that Ψ max is 1/5-power proper to Ra as Ra > , between is the transition regime. (b) - - -600 0.2 0.4 0.6 0.8 100.020.040.060.080.1 (a) Figure 2-16: The flow stream function (a) and temperature field (b) of Ra = 10 . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positive and negativevalues, respectively. 41 hapter 2 Horizontal Convection The power law of Ra at high Rayleigh number is,Ψ max = 0 . Ra / . (2-5)Similar to the numerical simulation by Siggers et al (2004), Ψ max ∼ Ra / to Ψ max ∼ Ra / as10 < Ra < . But there is an obvious 1/5-power law as Rayleigh number is higher enough, whichimplies that this 1/5-power law is valid only at relatively higher Ra. The power law obtained bySiggers et al (2004) is only valid for transition regime.Second, the power law for heat flux is also studied. As in Fig.2-17b, the heat flux is seldomchanged as the increase of Ra. In fact, the heat flux is dominated by heat conduction in this regime[48, 56]. However, this changes to convection dominance as Ra larger than a critical value Ra ≃ ,and N u increases quickly as Ra . Then, a 1/5-power law emerges as Ra > , N u = 0 . Ra / . (2-6)Similar to the numerical simulation by Siggers et al (2004), there is no 1 / / N u ∼ Ra / max and N u , there are always three regimes: linearregime, transition regime and 1/5-power regime. However, the transition from one regime to an otheris not synchronous for Ψ max and
N u , where the transition for heat flux is leading than for the flowstreamfunction. § In summary, the horizontal convection at high Rayleigh number in a rectangle cavity with aspectratio of 1 : 10 is numerically simulated. According to the results within the regime of 10 < Ra < , three continues regimes are obtained: linear regime ( 10 < Ra < ), transition regime( 10 < Ra < ) and 1/5-power law regime ( 10 < Ra < ). For the flow strength, a 1/3-power Ra -1 Ψ max Ra Nu Figure 2-17: The flow stream function Ψ max (a) and heat flux (b) vs. Ra.42 law of Ra is fitted when Ra is not high enough ( 10 < Ra < ). However, a 1/5-power law isobtained as Ra is high enough ( 10 < Ra < ). The 1/5-power law confirms Rossby’s analysisand implies that 1/3-power law of Ra for Nusselt number by Siggers et al. is over estimation. § § Horizontal convection, in which the water is unevenly heated at the horizontal surface, was takenas a model of abyssal ocean circulation. As the abyssal ocean circulation plays an important role inclimate change, the horizontal convection has intensively been explored in recent years [51, 53, 54]. Itcan be set to motion by any small temperature gradient, unlike the Rayleigh-B´enard convection. Butsimilar to Rayleigh-B´enard convection, the horizontal convection may be unsteady at high Rayleighnumbers Ra . There is a critical Rayleigh number Ra c , and the steady flow is unstable and becomesunsteady when Ra > Ra c . The unsteady flow in horizontal convection was first found by numericalsimulation [51], then was observed in the experiment at Ra > [53]. This unsteady flow is provedto be non-turbulent even as Ra → ∞ , though the flow field seems to be chaotic [51]. The investigationon the unsteady horizontal convection flow is relatively less, except for [51, 53, 67]. However, theyhave mainly focused on how the turbulent plume maintains a stable stratified circulation. Yet howthe horizontal convection turned to be unsteady remains an elusive problem.To understand this problem, both Ra c for the onset of unsteady flow and instability mechanismare of vital. Paparella and Young [51] found Ra c ≈ × at Pr = 1 in their simulations, which issignificantly smaller than others’ results. For example, Rossby (1965), Wang and Huang (2005) foundthe flow is steady and stable for Ra < × in their experiments [52, 54]. Yet some other numericalsimulations [55, 56, 62] have not found unsteady flows for Ra < . Paparella and Young [51]explained this difference as: (i) lower aspect ratio ( H/L = 1 / Ra < even at a much lower aspectratio ( H/L = 1 /
10 ) [68]. Thus, it maybe the middle plume forcing that leads to destabilization atlower Rayleigh numbers.Our interest here is to verify their second hypotheses. Is the flow with middle plume forcing lessstable than the sidewall plume forcing? How the instability occurs? To investigate these problems,more accurate numerical prediction of Ra c is need for both forcing cases, for the spatial resolution ofsimulation is very coarse used (e.g. 128 ×
32 meshes are used in [51]). Then the flow field under bothmiddle and sidewall plume forcings are compared, which leads to an affirmative answer of the aboveproblem.Similar to the previous investigations, we consider the horizontal convection flows within thetwo-dimensional domain, and the Boussinesq approximation is assumed to be valid for these flows.As shown in Fig.2-18, the horizontal (y) and vertical (z) regimes are 0 ≤ y ≤ L and 0 ≤ z ≤ H ,respectively. Similar to [52], the depth L is taken as reference length scale and A = H/L = 1 / ω are introduced. The velocity −→ u = ( v, w ) , where horizontal velocity v = ∂ Ψ ∂z and vertical velocity w = − ∂ Ψ ∂y , respectively. Thegoverning equations in vorticity-streamfunction formulation are [48, 51, 56]:43 hapter 2 Horizontal Convection ∂T∂t + J (Ψ , T ) = ( ∂ T∂y + ∂ T∂z ) (2-7a) ∂ω∂t + J (Ψ , ω ) = − Pr( ∇ ω + Ra ∂T∂y ) (2-7b) ∇ Ψ = − ω (2-7c)where J (Ψ , φ ) = ∂ Ψ ∂y ∂φ∂z − ∂φ∂y ∂ Ψ ∂z denotes the nonlinear advection term. There are two important di-mensionless parameter in Eq.(2-7), i.e. Rayleigh number Ra = α T ∆ T gL / ( κν ) and Prandtl numberPr = ν/κ , where g , α T , ∆ T , L , κ and ν are gravity acceleration, thermal expansion coefficient,surface temperature difference, length of horizontal domain, thermal diffusivity and kinematic vis-cosity, respectively. Alternatively, Paparella and Youngs used vertical length H as length scale, so Ra = 64 Ra H , where Ra H is the vertical Rayleigh number by using vertical length H as unit [51].More specifically, we consider the horizontal convection in a rectangle tank at Pr = 1 . The tankhas same velocity boundary condition as that in [51], i.e. free slip and no stress at the walls. Inaddition, two different surface forcings are used, which are central symmetric. One is middle plumeforcing as T = [1 + cos(2 πy )] / T = [1 − cos(2 πy )] / T = cos( πy/
2) [55], there are two symmetric cells inthe flow field under such forcings (e.g. Fig.2-18 and Fig.2-24), when the flow is symmetrically steadyand stable. In addition, the middle plume forcing in the left cell is the same with the sidewall plumeforcing in the right cell (see Fig.2-18 behind). Thus in the steady flows, both forcings will lead to thesame flow patterns except for a position shift, which is proved by the following investigation.There are two important quantity describing the circulation, i.e. the non-dimensional streamfunc-tion maximum and the non-dimensional heat flux. The non-dimensional streamfunction maximumΨ max = Ψ ∗ max /ν , where Ψ ∗ max is the maximum of the dimensional streamfunction.The above Eq.(2-7) is solved with finite different method in non-uniform grids. Crank-Nicholsonscheme and Arakawa scheme [e.g. 60, 61] are applied to discretize the linear and nonlinear terms,respectively. Comparing to the other schemes, Arakawa scheme is more accuract but more expensive,and it has also been applied to horizontal convection flows at high Rayleigh number [62, 68]. Table2-1 shows the validation of the scheme with nature convection problem. A fine spatial resolution meshof 512 ×
128 is used to eliminate numerical instability. § First, the middle plume forcing is considered, which is steady and stable for
Ra < . × .Fig.2-18 shows the flow field (a) and temperature field (b) of Ra = 5 × with Ψ max = 59 .
83 , inwhich the flow is symmetric, steady and stable. In this case, the center line symmetrically separatesthe flow field into two parts, like a free slip wall. There is a vigorous downward jet in the center oftank corresponding to the middle plume forcing (Fig.2-18a), where the vertical velocity field has aminimum of w = − - - -30-40-50 y z (a) . . y z (b) Figure 2-18: The flow stream function (a) and temperature field (b) of Ra = 5 × . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positive and negativevalues, respectively.temperature field is very simple. An obvious boundary layer exists near the surface in temperaturefield, which leads to a 1/5-power law of Ra for heat flux [e.g. 48, 52, 56]. And below the temperatureboundary layer, the temperature is almost homogeneous due to the convection. Thus there is a verystrong stratification near the surface ( ∂T /∂z ∼ Ra / ) but a very weak stratification in other region( ∂T /∂z ∼ × , which is significantly larger than the value obtained before [51].To find the critical Rayleigh number Ra c , the growth rate of perturbation φ ( t ) is calculatednumerically. And φ ( t ) is assumed to satisfy φ ( t ) = e σt φ (0) , where σ = σ r + iσ i is the complex growthrate of disturbance. It is found that the onset of unsteady flow is at Ra c = 5 . × , as shown inFig.2-20. For Ra = 5 . × , the flow is stable and the growth rate is approximately σ r = − .
12 .But the flow is unstable and the growth rate is approximately σ r = 0 .
03 for Ra = 5 . × . Thusthe critical Rayleigh number Ra c is obtained 5 . × < Ra c < . × . The accurate valueof Ra c = 5 . × is obtained by interpolating from the above result. Moreover, the onset ofunsteady flow is found to occur via Hopf bifurcation. As Fig.2-20 shows, the image part of growth
500 -1000 -
00 1000 y z (a) - y z (b) Figure 2-19: The horizontal (a) and vertical (b) velocity fields of Ra = 5 × . It is steady andstable and symmetric with middle plume forcing, solid and dashed curves for positive and negativevalues, respectively. 45 hapter 2 Horizontal Convection rate is nonzero and the eigenmode of perturbation is periodic. This Hopf bifurcation of the horizontalconvection has not seen reported yet, and privous investigations dealt only with chaotic flows.Meanwhile, the evolution of the perturbational vorticity fields during the fist half period at t = 0(a), t = T / t = T / t = 3 T / Ra = 5 . × are depicted in Fig.2-21, respectively.The perturbational vorticity fields are symmetric about centerline, which implies that the horizontalvelocity is nonzero at centerline. It can be seen that the perturbation tripole A (the shadowed ellipsein Fig.2-21a) is generated from central downward jet, then propagates and amplifies along the centraljet downward to the bottom wall (Fig.2-21b,c,d). When tripole A approaching to the bottom, itbecomes weaker and weaker and breaks into two parts: the left and the right near the bottom, whichcan be seen from the evolution of tripole B (the shadowed rectangle in Fig.2-21a). And the mean flowadvects the broken vortexs horizontally along the bottom wall (Fig.2-21b,c,d). Then in the secondhalf period, a reverse tripole will generate right the same place of vortexes A at t = T / Ra = 5 × , in which the flow is symmetric, steady and stable like that under middle plumeforcing. There are two strong downward jets near the walls corresponding to the sidewall plumeforcing (Fig.2-24a). As mentioned above, the sidewall plume forcing will lead to exactly the same flowpattern as the middle plume forcing does except for a position shift, which can be seen from Fig.2-18and Fig.2-24. As the flow is stable, the center line like a free slip wall symmetrically separates thetwo cells. The left cell in Fig.2-18 is exactly the same with the right cell in Fig.2-24. However, the Ra σ i σ r Figure 2-20: Growth rate σ r (solid) and σ i (dashed) vs. Ra , respectively.46 z (c) y z (d) y z (a) y z (b) y Figure 2-21: The perturbational vorticity fields at t = 0 (a), t = T / t = T / t = 3 T / Ra = 5 . × , solid and dashed curves for positive and negative values, respectively. z (c) y z (d) y z (a) y z (b) y Figure 2-22: The perturbational vorticity fields at t = T / t = 5 T / t = 6 T / t = 7 T / Ra = 5 . × , solid and dashed curves for positive and negative values, respectively.47 hapter 2 Horizontal Convection y z Figure 2-23: The vorticity of Ra = 5 . × with vertical velocity w (shadowed as w > Ra c ≈ . × (with 768 ∗
192 meshes) in this case. As noted above, the flow is much more stablewith the sidewall plume forcing than that with the middle plume forcing, though both forcings leadto the same flow patterns. This is very interesting, and can be understood from the mechanism ofinstability.It’s found that the rigid wall suppresses the perturbation, which leads a more stable flow with thesidewall plume forcing than that with the middle plume forcing. As the flow loss stability is due tostrong velocity shear in the center in horizontal convection, the smaller of the shear the more stable ofthe flow. In the case of middle plume forcing, the perturbation with nonzero horizontal velocity occursat strongest downward jet. And the perturbed flows cross the center line and propagate downstream.However, in the case of sidewall plume forcing, these crossing flows are suppressed by rigid walls. Sothat the critical Rayleigh number is much larger in this case. Paparella and Young (2002) hypothesizedthat middle plume forcing may lead to a destabilization of the flow. Here this hypotheses is provedboth physically and numerically. § In conclusion, the onset of unsteady flow is found to occur via a Hopf bifurcation in the regimeof
Ra > Ra c = 5 . × for the middle plume forcing at P r = 1 , which is much larger thanthe previously obtained value. Besides, the onset of unsteady flow is due to shear instability of . . y z (b) -20-10 -30-50 -40 y z (a) Figure 2-24: The flow field (a) and temperature field (b) of Ra = 5 × , which are steady and stableand symmetric with sidewall plume forcing. 48 central downward jet. Finally, the second hypotheses of Paparella and Young (2002) for instabilityis numerically approved, i.e. the middle plume forcing can lead to a destabilization of the flow atrelatively lower Rayleigh numbers. Acknowledgements
This work was original from author’s dream of understanding the mechanism of instability in theyear 2000, when the author was a graduated student and learned the course of hydrodynamic stabilityby Prof. Yin X-Y at USTC (China). The author thanks Prof. Sun D-J at USTC (China), Dr. YueP-T at UBC (Canada) for their help on preparing the report.The support of NSFC (No. 40705027 and No. 10602056) and the National Science Foundation forPost-doctoral Scientists of China are gratefully acknowledged. The author would like to acknowledgeProfessor Wang W at OUC (China) and Professor Huang R. X. at WHOI (USA) for the usefulcomments and suggestions on the studies of horizontal convection during the preparation of the secondpart of the report. 49 hapter 2 Horizontal Convection ibliography [1] Lord Rayleigh. On the stability or instability of certain fluid motions. Proc London Math Soc.1880, :57–70[2] W. O. Criminale, T. L. Jackson, R. D. Joslin. 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Nu (b) y z ( a ) y z ( b ) a10 Ψ v (a)
50 100 150 20010 N stableunstable Ra ( b ) a10 DNS0.48Ra
Nu (b) a T=sin( π y/2)T=y Nu Nu=0.475Ra
Nu=0.62Ra (b) a Ra Ψ pmax ( a ) Ra Ra ( b ) Nu a π y/2)T=y Nu ( b ) Ra π y/2)T=y Ψ κ max ( a ) a10
22 DNS0.325Ra Ψ v (a) a T=sin( π y/2)T=y Ψ κ (a) Ω p full-penetratingpartical-penetrating heatflux z Ω f (a) ω i (b) (b) -3 ω × f T .001 -0.001 - . z (a) . -0.002 -0.002y z (b) f T N=40N=64N=80 ( a ) Ψ max z (b) y z (a) - -500 - y z (b) - - -1500 y z (a) -2000 0 5 y z (a) y z (b) z (b) y z (a) z (a) y z (b) z (b) y z (a) y z z (a) y z (b) y z (a) . . y z (b) (b) y z (c) y z (a) y a σ r -0.0004-0.000200.00020.0004 (a) φ t -0.001-0.000500.00050.001 (b) φ t (b) y z (c) y z (a) y q y z Ra=1Ra=1E8Ra=2E8Ra=5E8 q y z z (a) z (b) y z (a) y z (b) z (a) Ra=2*10 Ra=5*10 Ra=10 q Z (b) y z (a) . . . y z (b) z (a) z (b)
20 0 y z ( b ) y z ( c ) y z ( d )
520 0 y z ( a ) .20740.190.1730.162 . y ( a ) z . . y ( b ) z . . . y ( c ) z . . y ( d ) z a p Ra p Ψ max ( a ) Ra P PPP0.0. ( b ) r c i U min U max Howard’s semicircle
Triangle axmin
UUUc r k r c k (a) AB r c k (b) ABC k N ω i ξξ