Blue Holes and Froude Horizons: Circular Shallow Water Profiles for Astrophysical Analogs
BBlue Holes and Froude Horizons:Circular Shallow Water Profiles for Astrophysical Analogs
Amilcare Porporato ∗ Department of Civil and Environmental Engineering and High Meadows Environmental Institute,Princeton University, Princeton, New Jersey 08540, USA
Luca Ridolfi † Department of Environment, Land and Infrastructure Engineering,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
Lamberto Rondoni ‡ Dipartimento di Scienze Matematiche, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy andINFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy (Dated: January 19, 2021)Interesting analogies between shallow water dynamics and astrophysical phenomena have offeredvaluable insight from both the theoretical and experimental point of view. To help organize theseefforts, here we analyze systematically the hydrodynamic properties of backwater profiles of theshallow water equations with 2D radial symmetry. In contrast to the more familiar 1D case typicalof hydraulics, even in isentropic conditions, a solution with minimum-radius horizon for the flowemerges, similar to the black hole and white hole horizons, where the critical conditions of unitaryFroude number provide a unidirectional barrier for surface waves. Beyond these time-reversiblesolutions, a greater variety of cases arises, when allowing for dissipation by turbulent friction andshock waves (i.e., hydraulic jumps) for both convergent and divergent flows. The resulting taxonomyof the base-flow cases may serve as a starting point for a more systematic analysis of higher-ordereffects linked, e.g., to wave propagation and instabilities, capillarity, variable bed slope, and rotation. “(l’Idraulica), sempre da me tenuta per difficilissima e piena di oscurit`a.”Galileo Galilei [letter to R. Staccoli, January 16, 1630]
I. INTRODUCTION
Browsing the recent literature on fluid analogies in as-trophysics [1, 2], Galileo may object that his quote aboutthe ’obscurities of hydraulics’ actually does not refer tothe darkness of the black holes, but to the mysteries offluid dynamics, which still persist today. Indeed, it wasnot until the early 1970’s, that striking analogies betweenthe properties of black holes and thermodynamics werepointed out, associating the area of a black hole eventhorizon with the thermodynamic entropy. A kind of sec-ond law was proven to hold [3–6] (cf. Ref. [7] for a review)and a correspondence with hydrodynamic systems wasobserved [8]. Whether such analogies are to be taken asexpressions of bona fide thermodynamic properties wasan issue at the time, and so remains today [9–11]. ∗ [email protected] † luca.ridolfi@polito.it ‡ [email protected] In 1981, Unruh [1] noted that there exist well under-stood acoustic phenomena, reproducible in the labora-tory, which formally enjoy the same properties of blackholes, as far as the quantum thermal radiation is con-cerned (the acoustic metric is conformal, but not identi-cal, to the Painlev´e–Gullstrand form of the Schwarzschildgeometry [2]). Independently, acoustic analogies wereconsidered also by other authors, notably by Moncrief[12] and Matarrese [13]. In particular, Unruh’s work im-plies that exotic phenomena, such as black hole evapora-tion, may be tested in controllable experiments on Earth!The way was paved for a research line now known as ana-logue gravity [2], which has led to numerous experimentalanalogues of cosmological gravitational phenomena.Most notable for their beauty and variety are the hy-draulic experiments, which in turn are connected by anal-ogy to polytropic gasdynamics [14, 15]. For instance, Ref.[16] reports on a shallow water experiment mimicking thedevelopment of a shock instability in the collapse of a stel-lar core, that makes highly non-spherical the birth of aneutron star, in spite of the underlying spherical sym-metry. The element of interest here is the 2-dimensional a r X i v : . [ phy s i c s . f l u - dyn ] J a n hydraulic jump, in a convergent radial flow, which corre-sponds to the accretion spherical shock that arises abovethe surface of the neutron star when it is being generated.The diverging radial case is instead associated to a whitehole, a kind of time reversed black hole [17, 18], which isexperimentally realized even with superfluids [19, 20]. InRef. [21], the scattering of surface waves on an acceler-ating transcritical 1-dimensional flow are investigated, ascorresponding to the space time of a black hole horizon,while in Ref. [22], gravity waves have been used to inves-tigate instabilities of both black and white holes. Theseare but a few examples (see also [23–25]) out of the manyone can find in the specialized literature.Using the same mathematical framework to describevery diverse physical situations has repeatedly provensuccessful. Laplace considered Newtonian mechanicsequally suitable to describe the ’movements of the great-est bodies of the universe and those of the tiniest atom’ [26]. Classical mechanics is effective not only whendealing with several macroscopic objects, under non-relativistic conditions, but it also constitutes the basisof statistical mechanics, which deals with huge numbersof atoms and molecules. The planets orbiting the Sunand the molecules in a glass of water are separated by 52orders of magnitude in mass, and by 23 orders of mag-nitude in numbers. The shallow water experiment andthe collapse leading to a neutron star [16] are separated,after all, by only six orders of magnitude in size.Within this context, our contribution aims at provid-ing a systematic classification of the so-called backwaterprofiles (i.e., the elevation of the free surface as a func-tion of the streamwise coordinate), possibly connectedby shock waves (i.e., the hydraulic jumps), as solutionsof the shallow water equations in circular symmetry. Asimple but systematic discussion of this type may havethe merit of connecting apparently different flow configu-rations. While most of the shallow water literature refersto 1D streams, here we focus on the role of the circu-lar symmetry enforced by the continuity equation, whichopens the possibility for convergent or divergent flows; wealso pay attention to the role of dissipation, provided notonly by friction but also, when present, by shock wavesin the form of hydraulic jumps.The astrophysics literature has mainly dealt with caseswith circular hydraulic jumps, either convergent [16] ordivergent [17, 18], as well as with 1D currents with criticaltransitions over obstacles [23, 24]. While these dissipativecases are characterized by strong energy losses due to thepresence of hydraulic jumps, and therefore are not sym-metric with respect to time or velocity inversion, here wealso emphasize the presence of an inviscid solution, whichobeys this symmetry. The corresponding analytical so-lution is of particular interest, as it represents a closeanalog of the black hole; herein, the subcritical (i.e., sub-luminal, in the analogy) convergent current acceleratestowards a ’Froude horizon’, where the velocity becomescritical (i.e., equal to the speed of the surface waves).The white hole analogy emerges naturally, when the flow velocity is reversed.Turbulent friction, acting in the direction opposite tothe flow, allows for the appearance of a critical pointin the dynamical system describing the free surface pro-files, and thus it indirectly allows for the presence ofstable hydraulic jumps, represented as sharp discontinu-ities in the surface profiles. Depending on the boundaryconditions, both converging and diverging cases with nojumps (called here dissipative black and white holes, re-spectively) are also found, along with the correspondingcases with shock waves.We confine our discussion to steady state profiles, inboth inviscid and turbulent conditions, noticing howeverthat the laminar case presents no qualitative differences.While the interesting effects of capillarity and rotationare left for future work, we hope that the present analysismay be useful to provide a classification of base flows toanalyze systematically the links between shallow waterprofiles and their astrophysical analogs. II. GOVERNING EQUATIONS
The starting point of the analogy are the well-knownshallow water equations [14, 15], namely the continuityequation ∂ t h = − ∇ · ( h v ) , (1)where h is the water depth, and v is the depth-averagedvelocity, which obeys the momentum equation ∂ t v + v · ∇ v + g ∇ h = ∇ z b − j , (2)where g is the gravitational acceleration, considered con-stant, z b is the bed elevation, and j is the frictional force.Friction is modeled as j = | v | α C h β v , (3)where C is the Chezy’s coefficient, which is inverselyproportional to the friction coefficient.In what follows, we will only consider horizontal streambeds, ∇ z b = 0, and will focus on the case of α = 1 and β = 1, typical of fully developed turbulent flows. Forsimplicity, we assume C to be constant, according toBresse’s hypothesis [27]. More complicated formulations[28–31], including the laminar case with α = 0, β = 2[32]), do not change the picture qualitatively. Other ef-fects, such as rotation and capillarity, although poten-tially very interesting, are neglected here. We will returnto a discussion of their effects and to possible extensionsof this work in the concluding section. III. ISENTROPIC CASE
When reduced to a circularly symmetric problem, insteady state and in the absence of friction, the continuity
FIG. 1. Isentropic (inviscid) case. Top left panel: stream profiles as given by Eq. (8); the red point marks the critical condition ξ min = 3 √ / y ( ξ min ) = 2 /
3. Top right panel: behavior of the velocity (dashed lines) and Froude number (solid lines) in thesubcritical (black lines) and supercritical (red lines) conditions. Bottom panel: 3D view of the dimensionless water-level profilefor the subcritical branch of the inviscid solution (8), corresponding to the black hole. A qualitative rendering of the ’lightcones’ for the propagation of small waves is also shown. equation becomes 2 πrvh = Q, (4)where Q is the total volumetric flowrate, while the mo-mentum equation takes the form ddr (cid:18) h + v g (cid:19) = 0 , (5)and it can be integrated as h + v g = H, (6)where H is the constant head (energy per unit weight offluid).Eliminating the velocity from the previous equations, h + Q g (2 πr ) h = H, (7)consistently with [32–34], suggests a normalization with y = h/H and ξ = rR , where R = Q πH √ gH is the radius at which the given discharge passes with Torricellian ve-locity √ gH and height H . As a result, ξ = 1 y √ − y . (8)where, by definition, one has ξ > y ∈ (0 , ν = 1 ξy . (9)Note that ξ min = √ and y ( ξ = ξ min ) = (see Eq. (40)in [22]). Each branch of the solution represents two possi-bilities, since the flow direction can be inverted, becausethe equations are invariant under changes in the flow di-rection. In the astrophysical analogy, the top branchrepresents a black hole, when the flow is convergent, anda white hole, when the flow is divergent, both with nodissipation.The occurrence of a minimum radius, ξ min , can beunderstood by considering the radian-specific discharge, Q u = Q/ πr , or q u = 1 /ξ in dimensionless form, whichcombined with the stream profile (8) yields q u = y (cid:112) − y (10)showing that q u = q u ( y ) is a non-monotonic relation, witha maximum at y = 2 /
3, corresponding to q u = 1 /ξ min .This limit behavior arises because a stream approach-ing smaller radii (flowing according to one of the twobranches) can carry that flow per radian only until themaximum value of q u is attained, and the stream reachesthe depth y = 2 / ξ min . For smaller radii, the streamcannot carry that much energy H and discharge Q , whileconserving them. The condition of maximum q u is called critical (i.e., y = 2 / ≡ y c ).By introducing the Froude number F r = v √ gh = 1 ξ (cid:114) y , (11)the critical condition corresponds to F r = 1, which isplotted as the red line y c = ( √ /ξ ) in Fig. 1. Accord-ingly, the upper (lower) branches showed in Fig. 1 arecalled subcritical (supercritical) conditions. Notice thatthe Froude number is the ratio between the stream ve-locity, v , and the propagation celerity of small-amplitudesurface waves, √ gh (first obtained by Lagrange [35]).It follows that subcritical streams are characterized bysurface waves that can propagate against the stream( F r <
F r > ξ min corresponds to thethreshold at which surface waves are no longer able togo up the current. As a subcritical current flowing to-wards the central hole with decreasing water depth h , thestream velocity v increases, while the wave celerity √ gh decreases. It follows that the Froude number graduallyapproaches 1, and the stream reaches this critical valueprecisely at a ’Froude horizon’, inside which the surfacewaves are no longer able to go upstream, namely to runaway from the hole. In this sense, the surface waves re-semble the light in the Schwarzschild problem. However,while in the black hole the velocity of the falling observerdecreases proportionally to the square root of the radiusand the light speed remains constant, here (see top-rightpanel in Fig. 1) the surface-wave speed changes in spacedepending on the water depth; as mentioned in the Intro-duction, the correspondence between the black hole andthe shallow water metrics is only conformal [2].Note that modifying the dependence of the bottomslope with the radius, the analogy could possibly be madetighter, as also briefly noted in [22] but this will not be pursued here. Another interesting point for futurework regards the conditions beyond the minimum radius,namely inside the Froude horizon, where different solu-tions, similar to the so-called interior Schwarzschild solu-tion [36], might account for the fact that the flow cannottake place with the same discharge and energy. IV. BACKWATER PROFILES DUE TOFRICTION
Allowing for the effects of turbulent friction, severaladditional configurations become possible. These can beobtained considering, in terms of water depth, the com-bined continuity and momentum equation as dHdr = ddr (cid:18) h + v g (cid:19) = − v | v | C h . (12)Using the normalization v/ √ gH = ν = 1 / ( ξy ) (where H is the stream head at the boundary) and the otherreference scales introduced before yields dydξ (cid:18) − ξ y (cid:19) = 2 ξ y − αξ y , (13)where α = sgn( v )2 g/C . Thus α is positive for turbu-lent flows proceeding along ξ (divergent) and negativefor flows that go against ξ (convergent). As a result, theslope of the water depth is dydξ = 2 y − αξξ y − ξ = N ( ξ, y, α ) D ( ξ, y ) . (14)Figure 2 shows the phase plots for some values of α .In these plots, three curves are highlighted: the profiles(8) corresponding to the inviscid case (black lines), andthe solutions of equations N ( ξ, y, α ) = 0 (green lines)and D ( ξ, y ) = 0 (red lines). Since the physical domain isbound to ξ >
0, green lines occur only if α > α = 0 case (i.e., no energy dissipation), the invis-cid solution reproduces the only possible stream profilescompatible with energy conservation: depending on theboundary condition, the subcritical or the supercriticalreach is selected (notice that the critical condition, wherethe reaches join, lies on the curve D ( ξ, y ) = 0). Dif-ferently, when dissipation occurs (i.e., α (cid:54) =0), the curverepresenting the flow starts from the boundary with en-ergy H and flows dissipating energy according to the Eq.(12). Since H was chosen as a reference energy to nor-malize the problem, it follows that the boundary condi-tion lies on the inviscid solution, where H ( ξ bc ) /H = 1),being ξ bc the radial position of the boundary. The re-maining part of the stream profile follows the curve ofthe phase space, departing from the boundary condition.In other words, the black curve corresponding to the so-lution (8) now becomes the locus of the initial conditionsof the backwater profiles. As expected, the phase trajec-tories are tangent to the inviscid solution only for α = 0.Becuase of dissipation, the flows are characterized by( h + v / g ) ≤ H , that in dimensionless form reads y + 1 ξ y ≤ . (15)Such a condition is satisfied only in the region of thephase space enclosed by the inviscid solution (8), thatis the area inside the black curve in Fig. 2. There-fore, physically meaningful current profiles correspond tophase lines in this region. The curve D ( ξ, y ) = 0 marksthe critical condition and separates the upper reach of thephase lines corresponding to subcritical streams ( F r <
F r > α . In thecase of α > ξ ), the phase space contains a focus where N = D = 0 – with coordinates { ξ f = (cid:112) /α , y f = (cid:112) α / } .The latter falls within the physically meaningful domainonly if α < / (9 √ γ h φr + βρ ( φrq ) v = const., (16)where the two addends refer to the hydrostatic and dy-namical force, respectively, γ is the fluid specific weight, φ is the central angle of the circular sector, β is the momen-tum coefficient, and ρ is the fluid density. In the previousrelation, bed friction and lateral hydrostatic componentshave been neglected.By introducing the dimensionless quantities and as-suming turbulent motion ( β (cid:39) y + 4 ξ y ≡ F ( ξ, y ) = const ., (17)namely the (dimensionless) specific force F that the twostreams have to balance immediately before ( F r >
1) andafter (
F r <
1) the hydraulic jump [37], i.e., F ( ξ j , y ) = F ( ξ j , y ), where the subscripts 1 and 2 refer to the super-critical and subcritical depths at the radial position, ξ j ,of the hydraulic jump. A minimum of the specific force, F = F min , occurs at ξ = (cid:112) /ξ , in correspondence tothe critical condition, D = 0.It is important to note that the hydraulic jump is astrongly dissipative phenomenon, related to the forma-tion of turbulence and vorticity. In dimensionless terms,the energy dissipation of the hydraulic jump can be cal-culated as the difference in total energy across the jump, [∆(∆ + 2 y ) / ( y y ξ j ) − ∆], where ∆ = ( y − y ) is thedepth difference across hydraulic jump.In terms of astrophysical analogs, the introduction offriction in the shallow water dynamics brings about ad-ditional cases, exemplified in Fig. 3: a neutron star(left column) and an analog of the dissipative white hole(right column); two other analogs are depicted in thesame figure (with dotted lines) and correspond to dissi-pative forms of the black and white holes. In the caseof the neutron-star analog, shown on the left of Fig. 3,the flows proceeds towards the center. Practically speak-ing, the fluid enters the domain from a circular sluicegate, placed along the external radius, and is drainedthrough a central hole (of dimensions larger than ξ min ).The flow is initially supercritical and becomes subcriticalafter a hydraulic jump. The central and bottom panelsshow that the Froude number exhibits a non-monotonicbehavior, as it first decreases in the supercritical reachand then increases when the stream becomes subcriti-cal. The reason for this lies in the hydraulic constraintthat the current must become critical at the edge of thecentral hole, as shown in the right central panel, wherethe stream reaches F r = 1 on the hole edge placed at ξ = 7 .
5. Accordingly, also the velocity shows a non-monotonic behaviour. Finally, the left lower panel high-lights that the hydraulic jump entails an abrupt energydissipation, which occurs where the specific forces of su-percritical and subcritical streams are equal.In the case of the dissipative white hole, illustrated inthe right panels of Fig. 3, the fluid flows along increasingvalues of ξ (e.g., as if coming from a vertical jet impingingthe bed close to ξ = 0) and it is initially in supercriti-cal conditions; a hydraulic jump then connects the pro-file to the subcritical one downstream. The central andbottom panels show that both the Froude number andthe stream velocity decrease monotonically along the ra-dius (although they would start increasing again, if theprofile were to be continued), with a step change at thehydraulic jump. The condition of equality of the specificforce dictates the radial position of the hydraulic jump,where a localized energy dissipation occurs. The subcrit-ical profile y ( ξ ) can be non-monotonic, since a maximumcan occur depending on whether the subcritical reach in-tersects or does not intersect the line N = 0, which isthe green line in Fig. 2.Fig. 3 reports also the profiles occurring when waterdrains or flows from a central hole or spring, withouthydraulic jumps. Such profiles, corresponding to dissipa-tive black holes, are characterized by subcritical streamsshown as dotted lines. In the drain case (left panels), theflow originates from an external circular reservoir, thenit accelerates converging towards the center and finallyenters the hole in critical conditions. In the spring case(right panels), a profile, analogous to a dissipative whitehole, starts from the critical condition ( F r = 1, wherewater emerges), gradually slows down and joins the sub-critical profile previously described in the case of whitehole with shock. ξ y ξ y ξ y ξ y FIG. 2. Phase space for α increasing in steps of 0.4, starting from -0.4 (top left). The red line is D = 0, while the green is N = 0; the black line is the solution of the inviscid case, Eq (8). These last two subcritical analogs with no hydraulicjumps are similar to those discussed in the isentropiccase. However, the presence of a maximum in the curve y = y ( ξ ) of the spring case, which does not occur in isen-tropic situations, highlights an interesting class of pro-files, connecting two critical horizons, that is peculiar tothe viscous case. An example of a white hole confinedbetween two horizons (see [22], Sec. XI) is shown in Fig.4, where the flow springs from critical conditions, reachesa maximum and then decreases returning to the criticalcondition before jumping off from the outer edge of cir-cular plate.The stability of the hydraulic jumps occurring in bothanalogs of Fig. 3 is an interesting matter. If one onlyconsiders the specific forces F , they both appear spa-tially stable: perturbations of their radial position areabsorbed by the consequent imbalance between the up-stream and downstream specific forces, so that eventuallythe jumps return to their original position. However, amore detailed momentum balance across the jump, thatincluded lateral hydrostatic pressures and bed friction,could alter this picture (see also [31, 38]), especially inconvergent cases [16], as in one-dimensional streams inconvergent or upward sloping channels, [39, 40]. Finally,it is worth mentioning that hydraulic jumps connectingsupercritical to subcritical streams are possible also inthe isentropic case. However, unlike the dissipative cases,their spatial position is undetermined, being marginallystable [41]. V. CONCLUSIONS
The solutions of the shallow water equations present avariety of configurations, which besides their direct fluiddynamic interest may also have useful implications asanalogs of specific astrophysical phenomena. For con-ditions of circular symmetry, the resulting steady statesolutions have been discussed with particular attention tothe transition between subcritical and supercritical con-ditions. The main cases are organized in Table I. Thesesteady state solutions may be realized in the laboratoryand can be used as base solutions to explore the modesof propagation of disturbances and instabilities.Starting from these configurations, several avenues forfuture research are suggested by the astrophysical analo-gies. Of particular interest is the stability of the hydraulicjumps. As already mentioned, this analysis is compli-cated by the presence of bottom friction and, in particu-lar, by the pressure forces along the circumference of theshock, whose quantification depends on the specific ge-ometry of the hydraulic jump [39, 40]. Moreover, theymay include oscillation and symmetry breaking instabil-ities [31, 38], including those nicely documented in theneutron star analogue [16].Along a similar line, one could conjecture the appear-ance of roll waves, i.e., pulsing and breaking waves, see[15, 42]), that could be realized in supercritical condi-tions with variable bottom slope. Apparently similarstar-pulsation phenomena are well known in the litera-ture [43–45]. In general, modifications of the bed slope Ξ (cid:144) H Ξ Ν , Fr 8 8.5 9 9.5 10 Ξ Ξ (cid:144) H Ξ Ν , Fr 4 5 6 7 8 9 10 Ξ FIG. 3. Dissipative solutions. The columns (see solid and dashed lines) refer to neutron star ( α = − .
1) and white hole( α = 0 .
1) cases, respectively. The first row shows stream profiles (solid lines; blue arrows indicate flow direction); the centralrow reports behaviors of velocity (dashed lines) and Froude number; the lower row displays behaviors of specific force (dashedlines) and stream head (referred to the initial stream energy, H ). In the lower row, the red and black lines refer to thesupercritical and subcritical streams, respectively. In all panels, dotted lines refer to profiles with no hydraulic jumps: thedrain case (left column) and the spring case (right column). (both downward and upward) introduce a degree of free-dom, which would allow for the interplay between energydissipation by friction and potential energy gain/loss towiden the gamut of hydraulic profiles and shock behav-iors.Finally, including rotations would be of interest forboth Kerr-Newman black holes and for exploring wave generation in vorticity-shock interactions [46, 47], whilecapillarity effects are known to generate lower wave-number disturbances in the upstream reach of obstacles[15], which have been linked to the Hawking radiation ofblack hole evaporation [24]. Extended thermodynamicformalism for turbulent flows, shocks and waves mightalso provide avenues to more concretely link black holeentropy to classical thermodynamics [48, 49]. Ξ FIG. 4. 2-horizon spring: dissipative, subcritical profile connecting two horizons ( α = 0 .
4, black line). Blue arrow indicatesflow direction, while red lines refer to the critical conditions (solid line) and Froude number along the profile (dashed line),respectively.TABLE I. List of shallow water analogs in circular symmetry (SUB=Subcritical flow; SUP=Supercritical flow; HJ=HydraulicJump; SFH=Smooth Froude Horizon).Shallow Water Astr. Analog Flow Dir. Flow Types Energetics Eq./Fig. Ref.Circular Jump Neutron Star Convergent SUP > HJ > SUB Dissipative Eq. (14), Fig. 3 [16]Drain TurbulentBlack Hole Convergent SUB > SFH Dissipative Eq. (14), Fig. 3Inviscid Drain Black Hole Convergent SUB > SFH Isentropic Eq. (8), Fig. 1 [22]Inviscid Spring White Hole Divergent SUB > SFH Isentropic Eq. (8), Fig. 1 [22]Spring TurbulentWhite Hole Divergent SUB > SFH Dissipative Eq. (14), Fig. 32-Horizon Spring ConfinedWhite Hole Divergent SFH > SUB > SFH Dissipative Eq. (14), Fig. 4 [22], Sect. XICircular Jump White Holewith Shock Divergent SUP > HJ > SUB Dissipative Eq. (14), Fig. 3 [18, 19]
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