Van der Waals Cascade in Supercritical Turbulence near a Critical Point

We investigate a quite strong turbulence in a supercritical fluid near a gas-liquid critical point. Specifically, we consider a case in which the Kolmogorov scale is much smaller than the equilibrium correlation length $\xi$. Although equilibrium critical fluctuations are destroyed by turbulence, $\xi$ still provides a crossover length scale between two types of energy cascade. At scales much larger than $\xi$, the Richardson cascade becomes dominant, whereas at scales much smaller than $\xi$, another type of cascade, which we call the van der Waals cascade, is induced by density fluctuations. Experimental conditions required to observe the van der Waals cascade are also discussed.

Introduction.-Nonlinearity, which appears ubiquitously in a broad range of phenomena, causes inevitable interference between widely separated time and space scales. One of the most extreme examples is found in fully developed turbulence. In turbulence, the kinetic energy is transferred conservatively and continuously from large to small scales in the so-called inertial range [1]. The mechanism of this remarkable phenomenon, the Richardson cascade, was intuitively explained by Richardson's depiction of a large vortex splitting into smaller vortices [2]. As a consequence of the energy transfer, the kinetic energy spectrum exhibits a power-law behaviorthe Kolmogorov spectrum [3,4]. The kinetic energy transported to small scales is dissipated at the Kolmogorov scale, where viscosity begins to predominate, so that the Richardson cascade is inevitably cut off at the Kolmogorov scale. The important point here is that in standard cases, the Kolmogorov scale is overwhelmingly larger than the microscopic length scales, such as the molecular mean free path [1]. Therefore, the cascade never reaches microscopic length scales in these cases.
Another notable instance in which nonlinearity causes strong interference between widely separated scales is found in critical phenomena. As an example, in the vicinity of a gas-liquid critical point, the correlation length of equilibrium density fluctuations ξ reaches a macroscopic order of magnitude [5,6]. We here consider the strong turbulent regime of a supercritical fluid near a critical point in which ξ is much larger than the Kolmogorov scale. Even for such strong turbulence, ξ still provides a length scale at which the stress induced by density fluctuations is comparable to the momentum flux. In this case, density fluctuations are driven by turbulence, so that the equilibrium critical fluctuations are destroyed. We then ask how the Richardson cascade is modified by density fluctuations in the turbulence near a critical point. Although turbulence in supercritical fluids has been studied over the past few decades, previous studies have focused on cases in which the Kolmogorov scale is larger than ξ [7][8][9][10].
We answer the above question by studying hydrodynamic equations including density fluctuations. Specifically, we include a density gradient contribution to the entropy functional to describe the effects of density fluctuations. Such a formulation that takes into account gradient contributions was originally proposed in the pioneering work of van der Waals [11], who introduced a gradient term in the Helmholtz free energy density to describe a gas-liquid interface, and the formulation has been widely used in statistical mechanics since the publication of seminal papers by Ginzbrug and Landau for type-I superconductors [12] and by Cahn and Hilliard for binary alloys [13]. Following the van der Waals theory, Korteweg proposed hydrodynamic equations that contain the van der Waals stress (vdW stress), arising from the density gradient [14,15], and Onuki generalized the theory by including the gradient contribution to both entropy and energy functionals [16,17].
In this study, we analyze the model using a phenomenological approach based on the Onsager "ideal turbulence" theory [18][19][20]. The Onsager theory describes the essence of turbulent behavior, such as the Richardson cascade and energy dissipation in the absence of viscosity, the so-called anomalous dissipation [21]. Although the Onsager theory involves sophisticated mathematical concepts such as weak solutions, it also provides a phenomenological perspective on the relation between cascades and the singularity of the velocity field. This theory has been recently extended to various turbulent phenomena, such as compressible turbulence [22][23][24][25][26][27] and plasma turbulence [28], and has also been intensively studied from a deep mathematical point of view related to convex integration [29][30][31][32][33].
In this Letter, we show that supercritical turbulence near a critical point exhibits the Richardson cascade and another type of cascade-the van der Waals cascade-induced by the van der Waals stress. First, we derive "Kolmogorov's 4/5-law," which states that the mean scale-to-scale kinetic energy flux becomes scaleindependent in the "inertial range." Second, we show the possibility of the existence of the van der Waals cascade. Furthermore, we consider the experimental conditions required to observe such van der Waals turbulence, which exhibits both the Richardson and van der Waals cascades.
Setup.-Let ρ be the mass density, v be the fluid velocity, and u be the internal energy density. For simplicity, we assume that a fluid is confined in a cube Ω = [0, L] 3 with periodic boundary conditions. We further assume that there is no vacuum region; i.e., ρ(x, t) > 0 for all x ∈ Ω and t ∈ R. Following the van der Waals theory, we include a gradient contribution to the entropy functional to describe enhanced density fluctuations near a critical point [11,16,17,34,35]: where ([u], [ρ]) := (u(x), ρ(x)) x∈Ω , s(u, ρ) denotes the entropy density, and c(ρ) ≤ 0 is the capillary coefficient. In the following discussion, we consider the case in which the capillary coefficient is a sufficiently smooth function of ρ; e.g., c(ρ) = const [17]. Through thermodynamic relations, the temperature T (u, ρ) and pressure tensor P are determined from (1) [36]: where p(u, ρ) denotes the pressure defined by s(u, ρ), I is the unit tensor, and Σ is the vdW stress tensor, which arises from the gradient contribution and is defined by (3) The time evolution of the densities of mass ρ, momentum ρv, and total energy ρ|v| 2 /2 + u is then governed by the Navier-Stokes-Korteweg equations [37][38][39]: where f denotes an external force acting at large scales ∼ L, λ is the thermal conductivity, and σ is the viscous stress tensor of the form Here µ and ζ are the shear and bulk viscosity coefficients, respectively. We assume that the viscous effect is sufficiently weak for the Kolmogorov scale to be sufficiently smaller than any other length scales. In the following, (4)-(6) are applied even to scales smaller than the equilibrium correlation length. Strictly speaking, dynamics at such scales should be described within the framework of fluctuating hydrodynamics [40]. In a turbulent regime, however, the equilibrium correlation may be cut off, and the noise terms may be irrelevant for energy transfer. We therefore assume that (4)-(6) are sufficient for our phenomenological argument.
Characteristic length scales.-Owing to the effect of the gradient contribution, several characteristic length scales that are not relevant in ordinary fluid turbulence become important. Let ρ 0 := ρ , c 0 := c(ρ 0 ), and T 0 := T be the typical density, capillary coefficient, and temperature, respectively, where · denotes the volume average Ω ·d 3 x/L 3 . In addition, let v 0 := (ρ 0 K T0 ) −1/2 be a velocity characterized by the isothermal compressibility K T0 := ρ −1 0 ∂ρ(T 0 , p)/∂p, which is zero at a critical point. One of the most crucial length scales is the correlation length of equilibrium density fluctuations which is expressed by the capillary coefficient c(ρ) and parameters in the entropy density s(u, ρ) [41]. The important point here is that even for strong turbulence, ξ still provides a characteristic length scale at which the vdW stress Σ and momentum flux ρvv are comparable.
Let c be such a length scale. Using an estimation that ρvv ∼ ρ 0 v 2 0 and Σ ∼ T 0 |c 0 |ρ 2 0 / 2 c , we obtain Note that Σ can be appreciable at small scales because it contains higher-order spatial derivatives. Therefore, at scales c , the effect of the vdW stress is small compared with the momentum flux, whereas at scales c , the vdW stress becomes relevant. This observation implies the possibility of the van der Waals cascade, induced by the vdW stress, at scales c . We attempt to seek other characteristic length scales by noting the local kinetic energy balance equation The first term on the right-hand side of (10), −p∇ · v, is the pressure-dilatation, which represents the conversion of kinetic energy into internal energy and vice versa. Recent numerical simulations [42,43] suggest that there is a characteristic length scale large such that the contribution to the global pressure-dilatation −p∇ · v from scales large is dominant, whereas the contribution from scales large is negligible. The second term on the right-hand side of (10), −Σ : ∇v, which we call the vdWstress-strain, arises because of the gradient contribution. It also represents the conversion between kinetic and internal energy. Because both the vdW stress Σ and strain ∇v change rapidly in space, there may be a characteristic length scale small such that the contribution to the global vdW-stress-strain −Σ : ∇v from scales small is negligible, whereas the contribution from scales small is dominant. In the following, we assume the existence of the intermediate asymptotic limit small large that satisfies small c and c large .
Main result.-Let Q flux be the scale-to-scale kinetic energy flux that represents the energy transfer from scales > to scales < : where Π is deformation work [44], which corresponds to the energy flux of the Richardson cascade, Λ (p) is baropycnal work [25,45], which arises because of compressibility, and Λ (Σ) is capillary work, which arises because of the gradient contribution. (Precise definitions are given below.) These three terms represent the energy transfer due to the momentum flux ρvv, pressure p, and vdW stress Σ, respectively. The first main claim of this Letter concerns the most crucial property of the energy cascade; i.e., in a steady state with a constant mean kinetic energy, Q flux becomes scale-independent in the "inertial range" small large : where eff := p∇ · v + v · f denotes the effective energy injection rate, which is scale-independent. We emphasize that, because Q flux can be expressed in terms of field increments v(x+r)−v(x) and ρ(x+r)−ρ(x), the relation (12) plays the same role as Kolmogorov's 4/5-law [4]. The second main result of this Letter is the prediction of the van der Waals cascade. In the range of c large , the Richardson cascade, induced by the momentum flux, becomes dominant, whereas in the range of small c , the van der Waals cascade, induced by vdW stress, develops: (13) Correspondingly, the velocity power spectrum exhibits a power-law behavior: Suggested experiments.-We consider the experimental conditions required for observing the van der Waals cascade. In the study of critical phenomena, CO 2 has been widely used because its critical state occurs under readily realized experimental conditions (T c = 304.13 K, p c = 7.3773 MPa, ρ c = 0.4678 g cm −3 ) [46,47]. In this case, the shear viscosity µ takes a value around 3.5×10 −4 g cm −1 s −1 [10,46,48]. We first estimate the Kolmogorov scale d , which can be estimated in terms of µ, ρ c , L, and v rms := |v| 2 as If we achieve a quite strong turbulent regime, in which Re ∼ 10 5 (e.g., v rms ∼ 100 m/s and L ∼ 0.1 m), the Kolmogorov scale is ∼ 800 Å. Therefore, if one can reach the vicinity of the critical point such that the correlation length is at least ∼ 10, 000 Å, it may be possible to verify our predictions by measuring the velocity field using hot-wire anemometry or laser Doppler velocimetry. To achieve a correlation length of that magnitude, we must control the system with an accuracy of at least [46,48]. Derivation of the main result.-We study the properties of kinetic energy transfer across scales using a coarsegraining approach that can resolve turbulent fields both in scale and in space. For any field a(x), we define a coarse-grained field at length scale as where G : Ω → [0, ∞) is a smooth symmetric function supported in the open unit ball with Ω G = 1, and G (r) := −3 G(r/ ) is the rescaling defined for each > 0. By coarse-graining (4) and (5), we can write the coarse-grained kinetic energy balance equation where we introduce the density-weighted coarse-grained velocityṽ := (ρv) /ρ to reduce the number of additional cumulant terms and to obtain a simple physical interpretation. Here, in :=ṽ ·f denotes the energy injection rate due to external stirring at scale , D := −∇ṽ :σ denotes the viscous dissipation acting at scale , and J represents the spatial transport of large-scale kinetic energy, which does not contribute to the energy transfer across scales. The first two terms on the right-hand side of (16), −p ∇ ·v and −Σ : ∇v , are the large-scale pressure-dilatation and vdW-stressstrain, respectively. Note that these two terms contain no modes at small scales < . Therefore, they contribute only to the conversion of the large-scale kinetic energy into internal energy and vice versa. The third term on the right-hand side of (16) denotes the scale-to-scale kinetic energy flux (11). The definitions and physical meanings of each term comprising Q flux are given as follows. Deformation work is defined by whereτ (v, v) := (vv) −ṽ ṽ , and it represents the work done by the large-scale (> ) strain ∇ṽ against the small-scale (< ) stressρ τ (v, v). Baropycnal work is defined by whereτ (ρ, v) := (ρv) −ρ v , and it represents the work done by the large-scale pressure gradient force −∇p /ρ against the small-scale mass fluxτ (ρ, v). Capillary work, which has a form similar to that of baropycnal work, represents the work done by the large-scale force ∇·Σ /ρ against the small-scale mass fluxτ (ρ, v). Note that in (16), only these three terms are capable of the direct transfer of kinetic energy across scales because each of the three terms has a form "large-scale (> ) quantity × small-scale (< ) quantity," whereas the other terms on the right-hand side of (16) do not.
In the steady state, the spatial averaging of (16) gives For the first term on the right-hand side, because p∇·v receives most of its contribution from scales large , it can be approximated as p ∇·v ≈ p∇·v for large . Similarly, because the contribution to Σ : ∇v from scales small is negligible, the second term becomes Σ : ∇v ≈ 0 for small . In addition, because the Kolmogorov scale is sufficiently smaller than other length scales and f acts at the large scale L, the viscous dissipation D and energy injection in can be approximated as D ≈ 0 and in ≈ v · f for small large , respectively [27]. Thus, in the intermediate scale range Note that although the mean total scale-to-scale kinetic energy flux Q flux is scale-independent in the inertial range, the three terms Π , Λ (p) , and Λ (Σ) are not necessarily scale-independent individually. In fact, because Σ is appreciable at scales c , Π and Λ (p) are dominant at scales c , whereas Λ (Σ) develops at scales c . Therefore, (21) can be further rewritten as We can also derive the result (22) more rigorously by evaluating the scale dependence of the energy fluxes using functional analysis [50].
We now consider the singularity of the velocity field that is necessary to satisfy (22). To this end, we investigate δa( ) := |a(x + ) − a(x)| for a field a(x) using the assumption of homogeneity and isotropy. The following pointwise estimation is based on a more sophisticated analysis using Besov spaces [50]. In the range of c large , the baropycnal work and deformation work are the main sources of the energy cascade. These two energy fluxes can be expressed in terms of increments: where we have usedρ ∼ ρ 0 assuming homogeneity. We have also used an estimation that ∇f ∼ δf ( )/ and τ (f, g) ∼ δf ( )δg( ), which can be made more rigorous using the L p -norm [50]. From this expression, it follows that Λ (p) → 0 as → 0 because the density increment is bounded from above as δρ( ) = O( ), which holds for all > 0 because the entropy functional contains the density gradient term ∝ |∇ρ| 2 < ∞. We therefore conclude that Π ≈ eff for c large . Then, from the expression (24), we obtain δv( ) ∼ ρ −1/3 0 1/3 eff 1/3 in this scale range. This result implies that the velocity power spectrum E v (k) follows the Kolmogorov spectrum: In the range of small c , the energy transfer is dominated by capillary work. The capillary work can be expressed in terms of increments: Here, note that we cannot naively estimate as in ∇·Σ ∼ δΣ( )/ because Σ already contains higher-order derivatives of ρ. In this scale range, we must consider the density increment δρ( ) because density fluctuations are appreciable. The scale dependence of δρ( ) can be complicated because of the strong turbulent effect [51], although it may be bounded from above as δρ( ) = O( ).
Here, as a first step to estimate the spectral exponent, we consider the consequence of imposing only the loose condition that δρ( ) = O( ). Then, by integrating by parts, we can estimate thatΣ δρ( ) ∼ Z, where Z is a scale-independent quantity [50]. Hence, we conclude that δv( ) ∼ Z −1 ρ 0 eff and E v (k) ∼ Z −2 ρ 2 0 2 eff k −3 in this scale range.
Concluding remarks.-In summary, we have shown that supercritical turbulence near a critical point can exhibit the van der Waals cascade. The interesting point here is that the results are similar to those reported for pure quantum turbulence [52]. This implies that pure quantum turbulence and van der Waals turbulence belong to the same "universality class." Therefore, our results may also provide an interesting perspective from which to understand quantum turbulence, which will help illuminate the role of quantized vortices and Kelvin waves. Finally, we remark that there is a possibility that the spectrum k −3 becomes shallower because of the depletion of nonlinearity [28,[52][53][54] or the regularity of the temperature and density gradient fields [50].
The problem addressed in this Letter could lead to an understanding not only of turbulence but also of the relation between the macroscopic and microscopic descriptions of nature. We therefore hope that experiments will be conducted to verify our predictions.
The present study was supported by JSPS KAKENHI Grant No. 20J20079, a Grant-in-Aid for JSPS Fellows, and JSPS KAKENHI (Nos. 17H01148, 19H05795, and 20K20425). Here, we derive the pressure tensor ( (2) and (3) in the main text). The equilibrium value of ([u], [ρ]) = (u(x), ρ(x)) x∈Ω in the isolated system enclosed by adiabatic walls, denoted as (u * (x), ρ * (x)), is determined as the maximizer of the entropy functional, It follows the conservation law, where E and N are constants. The variational equation is where λ 1 and λ 2 are Lagrange multipliers that are physically connected to the equilibrium values of temperature and chemical potential as λ 1 = 1/T eq and λ 2 = −µ eq /T eq , respectively. We defineμ such that the equilibrium condition is given by ∇μ = 0. We then determinep, such that ∇ ·p = 0 in equilibrium and p = p(u, ρ) when the gradient terms are ignored. To this end, we use a relation, which is derived from We first rewrite the second term on the right-hand side of (S7) in terms of the generalized chemical potential,μ, as Here, we used the relation, which follows from the definition ofμ (S6). By substituting this result into (S7), we obtain The equilibrium condition, ∇T = 0 and ∇μ = 0, leads to ∇ ·p = 0. In addition, it is evident thatp = pI when the density gradient is ignored. In the main text, we used the notation P =p to emphasize thatp is a second-order tensor and defined the van der Waals stress Σ as (S14)

Correlation length of equilibrium density fluctuations
In this section, we derive the correlation length of equilibrium density fluctuations and thus confirm that the correlation length is determined by the capillary coefficient and parameters in the entropy density. To this end, we introduce the Helmholtz free energy functional, where f := u − T s. Assuming small, slowly varying deviations in density, we consider the expansion of f in terms of the local deviation, δρ(x) := ρ(x) − ρ 0 , as follows: where K T is the isothermal compressibility, given by (S17) Substituting (S16) into (S15), we obtain where Here, the first power of δρ has been dropped considering the conservation of particles, and c(ρ) is replaced by c 0 := c(ρ 0 ) because the difference c(ρ) − c 0 is a higher-order contribution.
Introducing the Fourier transform of the density deviation, where V := L 3 and k ∈ (2π/L)Z, (S19) becomes and k := |k|. According to fluctuation theory in equilibrium statistical mechanics, δF (T, [ρ]) plays a role of an effective Hamiltonian describing density fluctuations of the system with temperature T . That is, the density correlation function takes the Ornstein-Zernike form [1], as follows: Here, ξ is the correlation length of density fluctuations where we introduce a velocity characterized by the isothermal compressibility, as follows: As an example, we consider a van der Waals fluid for which the equation of state is given as follows: where m denotes the mass of a particle; the heat capacity per unit volume is given by where a, b, and η are constants. In this case, the entropy density is given by where c is a constant. The critical density, temperature, and pressure are expressed as respectively. If ρ 0 = ρ c , the isothermal compressibility can be expressed as From (S23), (S27), (S28), and (S29), it is straightforward to confirm that the correlation length ξ is determined by the capillary coefficient c(ρ) and the parameters in the entropy density s(u, ρ).

Preliminaries
In this section, in preparation for the detailed derivation and explanation of the main result, we introduce the Besov regularity and investigate the scale dependence of energy fluxes and vdW-stress-strain.

a. Besov regularity
To investigate the scale dependence of the scale-to-scale kinetic energy fluxes, we assume that the following scaling laws hold for the absolute structure functions in the inviscid limit for p ∈ [1, ∞]: where A p is a dimensionless constant, · p := |·| p 1/p is the L p -norm, and δa(r; x) := a(x+r)−a(x) for any field a(x). The symbol ∼ denotes "asymptotically equivalence," i.e., f (x) ∼ g(x) for x → 0 if and only if lim x→0 f (x)/g(x) = 1.

b. Scale dependence of energy fluxes and vdW-stress-strain
In this section, we study the scale dependence of deformation work, baropycnal work, capillary work, and the vdW-stress-strain.
Deformation work. We now examine the scale dependence of the deformation work, Π = −ρ ∇ṽ :τ (v, v). Using the Cauchy-Schwarz and Hölder inequalities, we obtain where A p = |A| p 1/p for a matrix A = (a ij ) is defined using the Frobenius norm, that is, |∇ṽ | : For the second factor on the right-hand side of (S33), ∇ṽ p , if we use the relatioñ v =v +τ (ρ, v) ρ (S34) and the Minkowski inequality, we obtain Considering the first term on the right-hand side of (S35), it should be noted that, for any locally integrable function a(x), because d 3 r∇G(r) = 0. Subsequently, the triangle inequality gives Hence, where δa( ) p := sup |r|< δa(r; ·) p . For the second and last terms of (S35), using Propositions 3 and 4 in [8], we can obtain and Thus, combining the results (S38), (S39), and (S40), we obtain For the last factor on the right-hand side of (S33), τ (v, v) p/2 , if we use the relatioñ and the Minkowski inequality, we obtain Subsequently, using Proposition 3 in [8], we obtain Thus, from (S33), (S41), (S44), and condition (S30), we finally obtain as a rigorous upper bound. Note that the upper bound of (S45) becomes independent of in the case of σ p = 1/3. Baropycnal work. Next, we study the scale dependence of the baropycnal work, Λ (p) = (1/ρ )∇p ·τ (ρ, v). Using the Cauchy-Schwarz and Hölder inequalities, we obtain For ∇p p , from the inequality (S37), we obtain For τ (ρ, v) p/2 , using Proposition 3 in [8], we obtain From the requirements (S30), (S31), and (S32), we obtain This result implies that the mean baropycnal work, 1 , vanishes as O(( /L) σ3+σ p 3 ) for /L → 0. Therefore, the baropycnal work does not contribute to the transfer of kinetic energy across scales.
VdW-stress-strain. Finally, we investigate the scale dependence of the large-scale vdW-stress-strain −Σ : ∇v . From the Cauchy-Schwarz and Hölder inequalities, we obtain From a similar argument as (S53) and (S54), it follows that Therefore, using the inequality (S38), we obtain (S58)

Detailed derivation of "Kolmogorov's 4/5-law"
In the steady state, spatial averaging of the coarse-grained kinetic energy balance gives Next, we determine the scale range such that the right-hand side of (S59) becomes scale-independent. First, we can prove that the viscous dissipation term, D , can be ignored at scales that are much larger than the Kolmogorov scale, which is sufficiently smaller than other length scales [9]. In addition, because the external force, f , acts at the large scale L, it follows that [9] in := ṽ ·f Next, we show that p ∇ ·v ≈ p∇ · v for large . In the main text, large is introduced as the characteristic length scale such that the contribution to the global pressure-dilatation −p∇ · v from scales much larger than large is dominant, whereas the contribution from scales much smaller than large is negligible. The existence of such a characteristic length scale is ensured by the decay of the pressure-dilatation co-spectrum at a large k, which is well established for ordinary compressible turbulence [10,11]: where E (p) (k) is defined by Here, ∆k := 2π/L. Using the pressure-dilatation co-spectrum, the characteristic length scale large is explicitly defined, for instance, as From (S61) and (S63), it follows that the mean large-scale pressure-dilatation p ∇·v converges to the finite constant p∇ · v and becomes independent of at scales sufficiently smaller than large ; this is expressed as In the main text, small is introduced as the characteristic length scale, such that the contribution to the global vdW-stress-strain −Σ : ∇v from scales much larger than small is negligible whereas the contribution from scales much smaller than small is dominant. The existence of such a characteristic length scale is validated using (S58). The characteristic length scale small is explicitly defined, for instance, as where E (Σ) (k) is the vdW-stress-strain co-spectrum defined by From (S58) and (S65), it follows that the mean large-scale vdW-stress-strain, −Σ : ∇v ≤ |Σ : ∇v | = Σ : ∇v 1 , is negligible at scales sufficiently larger than small ; this is expressed as Combining these results, (S59) becomes Because Q flux can be expressed in terms of increments, as shown in Sec. 3 b, (S68) plays the same role as Kolmogorov's 4/5-law.

Existence of the van der Waals cascade
a. Basis of the estimationΣ δρ( ) ∼ Z Before we explain the existence of the van der Waals cascade using the results in Sec. 3 b, we explain the basis of the estimationΣ δρ( ) ∼ Z, which is used in the main text. From assumption (S31) and the estimation (S57), we obtain for all p ∈ [1, ∞]. This evaluation is the basis of the estimation,Σ δρ( ) ∼ Z.
b. Explanation of (22) Here, using the results in Sec. 3 b, we explain (22), i.e., From (S45) and (S55), it immediately follows that the upper bounds of the mean deformation work, Π , and mean capillary work, Λ (Σ) , have different dependences. In particular, in the case of σ 3 = 1/3, whereas in the case of σ 3 = 1, The "Kolmogorov's 4/5-law" states that the sum of the mean deformation work and mean capillary work, Π + Λ (Σ) , becomes scale-independent in the inertial range small large . From this law and the above observation, if we ignore the contribution of baropycnal work based on the evaluation (S49), it follows that a characteristic length scale λ exists such that the energy cascade due to the deformation work is dominant in λ large , whereas that due to capillary work is dominant in small λ (see Fig. S1). This is expressed as follows: From the definition of c , we expect that the crossover length scale λ is of the order of c . In fact, if we use an estimation that and Λ (Σ) where the symbol O denotes "same order of magnitude as," we obtain Thus, a two type of cascade occurs in the van der Waals turbulence, one in c large and the other in small c . The former is the Richardson cascade, which is induced by the deformation work, as in the case of ordinary turbulence. The latter is the van der Waals cascade, which is induced by capillary work, and its existence is specific to van der Waals turbulence.

c. Velocity power spectrum
Here, we explain the detailed derivation of the velocity power spectrum obtained in the main text. In compressible turbulence, we can consider the spectra of the velocity v and the density-weighted velocity, such as √ ρv [12]. In an ordinary compressible turbulence, high-resolution numerical simulations exhibit the Kolmogorov spectrum for both velocity [10] and density-weighted velocity power spectra [11] in the case where large is sufficiently larger than the Kolmogorov scale. In this subsection, we consider the spectra of both the velocity v and the density-weighted velocity √ ρv. Velocity power spectrum. First, we consider the pth-order (absolute) structure function for the velocity field, with assumed scaling exponent ζ p : where C p is a dimensionless constant. Using the Hölder inequality, it can be shown that ζ p is a concave function of p ∈ [0, ∞) [2,13]. From this property, it immediately follows that σ p = ζ p /p is a non-increasing function of p [13]. Note that the second-order structure function S v 2 ( ) ∝ ζ2 is related to the velocity spectrum E v (k) ∝ k −ζ2−1 , assuming isotropy.
Because σ 3 = 1/3 in c large and σ p is a non-increasing function of p, it follows that σ 2 ≥ 1/3 in this scale range. Hence, we can write ζ 2 = 2σ 2 ≡ 2/3 + µ/9, where µ is a positive constant. This additional constant µ corresponds to the so-called intermittency exponent [2]. Therefore, the velocity power spectrum exhibits the following asymptotic behavior: where C large is a positive constant. In small c , where the van der Waals cascade becomes dominant, we have seen that σ 3 = 1. Because σ p is a non-increasing function of p, it follows that σ 2 = 1. This result implies that the velocity power spectrum exhibits the following asymptotic behavior: where C small is a positive constant. This result is summarized in Fig. S2. Density-weighted velocity power spectrum. Next, we consider the spectrum of the density-weighted velocity √ ρv.
We consider the pth-order (absolute) structure function for the density-weighted velocity, with an assumed scaling exponent,ζ p .
whereC p is a dimensionless constant. Using the Hölder inequality, we can also see thatζ p is a concave function of p ∈ [0, ∞) [2,13]. Note that the second-order structure function, S √ ρv 2 ( ) ∝ ζ 2 , is also related to the density-weighted velocity spectrum, E(k) ∝ k −ζ2−1 , assuming isotropy. In this case, we cannot determine the exact value ofζ 2 because, from the mean value theorem, Minkowski inequality, and assumptions (S30) and (S31), δ( √ ρv)(r; ·) p ≤ B 1 δρ(r; ·) p + B 2 δv(r; ·) p ∼ B 3 v 0 |r| L σp as |r|/L → 0, where B 1 , B 2 , and B 3 are constants. Hence, and we cannot conclude thatζ p = pσ p in general. If we assume that ζ 2 ≈ζ 2 , as in an ordinary compressible turbulence, the asymptotic behavior of the kinetic energy spectrum E(k) can be obtained as whereC large andC small are positive constants. 6. Additional assumption on the temperature and density gradient field In this section, we show that, if we further assume the regularity of the temperature and density gradient field, the evaluation of the capillary work (S55) is not optimal, and the spectrum ∝ k −3 becomes shallower. To this end, we additionally assume the following: where σ ∇ρ p , σ T p ∈ [0, 1). Then, the evaluation ofΣ is modified as follows.