Proper orthogonal decomposition of eigen modes in a gas affected by a mass force
aa r X i v : . [ m a t h - ph ] A p r Proper orthogonal decomposition of eigen modes in a gasaffected by a mass force
Sergey Leble, Anna PerelomovaGdansk University of Technology,Faculty of Applied Physics and Mathematics,ul. Narutowicza 11/12, 80-233 Gdansk, Poland,[email protected], [email protected] 8, 2018
Summary
The relations connecting perturbations in acoustic and entropy modes in a gas affectedby a constant mass force, are derived. The background temperature of a gas may varyin the direction of an external mass force. The relations are independent on time. Theymake possible to decompose the total vector of perturbations into acoustic and non-acoustic parts uniquely at any instant. In order to do this, three quantities are required,according to the number of modes. In one dimension, the reference quantities may be totalperturbations in entropy, pressure and velocity. The total energy of flow is determined.The examples of dismemberment of the total field into acoustic and entropy parts relate tothe unperturbed temperature of a gas which linearly depends on the spacial co-ordinate.
Key words:
Atmospheric perturbations, Initialization of hydrodynamic field, Non-uniformmedia, Sound propagation, Heating.
PACS No.
In experiments of 1994 (Transarctic Acoustic Propagation Experiment, TAP), coherent acoustictransmissions via Arctic basin were studied. It was designed to monitor changes in Arctic Oceantemperature and sea ice properties. TAP demonstrated that the low frequency (19.6-Hz) signalspropagated with both sufficiently low loss and high phase stability to support the coherentpulse compression processing and the phase detection of the signals. These yield time delaymeasurements an order of magnitude better than what is required to detect the year changesin travel time caused by interannual and longer term changes in Arctic Ocean temperature.The TAP data provided propagation loss measurements to compare with the models to be usedfor correlating modal scattering losses with sea ice properties for ice monitoring. The traveltimes measurements indicated a temperature changes in the Arctic atmosphere, which has been1onfirmed by direct measurement from icebreakers and submarines, demonstrating the utilityof acoustic thermometry [1]. Climatological atmospheric velocity models predict infrasoundsignals from sources that occur at mid-northern latitudes. Such infrasound data have beenused to locate a bolid explosions in space and time. In [2] authors analyze travel time picksand use 3-D ray tracing to generate synthetic travel times based on various atmospheric modelsto show that the seismic network data instead reveal a predominant propagation direction. Asudden stratospheric warming event that reversed the zonal wind flow explains propagationproperties.A time-dependent, nonlinear, fully compressible, axisymmetric, f-plane, numerical model isused to simulate the propagation of acoustic waves in the mesosphere and thermosphere byintense deep convection in the troposphere [3]. The simulations show that major convectivestorms in the tropics launch acoustic waves into the mesosphere-thermosphere directly abovethe storm centers . The principal feature of the overhead acoustic wave field in the periodinterval of 3 to 5 min is a trapped oscillation below about 80 km altitude with a period of 5min and a nearly vertically propagating wave with about a 3-min period above this height.The vertically propagating thermospheric acoustic oscillations are waves propagating upwardfrom the thunderstorm source through the stratosphere-mesosphere. These predominantly 3-min waves are strongly driven for about 30 min after the storm event and weaken with timethereafter. The vertically propagating thermospheric acoustic waves may be the source of theF-region 3-min oscillations. Intense acoustic disturbances directly above thunderstorms mayalso be responsible for localized heating of the thermosphere [9].There is an important ingredient on which a direct problem of sound propagation or inverseproblems (such as acoustic thermometry) are based. It is a necessity the proper division of anatmospheric disturbance onto the characteristic types and hence a possibility to formulate aproblem of acoustic wave generation propagation and reflection as one of them. The theoreticaland numerical models as e.g. of [3] describing dynamics of gases and liquids affected by exter-nal forces are of great interest in geophysics, meteorology and wave theory [4]. The authorsbelieve that the theoretical models are more desirable than the numerical methods, which aretime-consumed and require large computer power and special converging and stable numeri-cal methods. Numerical methods, if necessary, are much more economical and successful, ifthey rely on the reasonable theoretical models. Moreover, some important problems may beresolved only theoretically. Some of these problems are considered in this study. Among other,they are: how many types of fluid motion exist, what they are and what energy associateswith every type of motion. The external forces make the background of a fluid non-uniform,with background density, temperature and pressure depending on spatial coordinates. Thatessentially complicates the definition of linear modes (motions of infinitely small amplitude)taking place in such non-uniform media [5]. The number of roots of dispersion equation, if it ispossible to determine them, or branches of possible types of motion, equals number of the con-servation equations [6],[4]. Even in the simplest case of flow in one dimension, the dispersionrelations may be introduced over all wave-length range only if the background pressure anddensity depend on the coordinate exponentially [1]. Anyway, there are three types of motionin one dimension: two acoustic branches and, if the thermal conduction of a fluid is ignored,the stationary (entropy, non-wave) mode. In the flows exceeding one dimension, the buoyancywaves appear and interact [4],[5].In this study, modes of one-dimensional flow are determined by independent on time rela-tions linking hydrodynamic perturbations. They are fixed for any mode and valid for arbitrary2ependence of the background temperature on co-ordinate. These relations give possibilityto distinguish modes analytically at any instant, to conclude about energy of everyone fromthem and to predict their dynamics. We will consider volumes of an ideal gas affected by aconstant mass force, with variable temperature of the background. The first results allowingto distinguish modes due to relations of specific perturbations, were obtained relatively to themotion of atmospheric gas affected by gravity in [5],[8]. Fluids different from ideal gases, in-cluding liquids, are briefly discussed in Concluding Remarks. It is undoubtedly of importancein applications of meteorology and atmospheric dynamics. The case of an ideal gas with con-stant temperature is considered in Sec.3. The examples relating to the linear dependence ofthe background temperature on vertical co-ordinate in the field of constant gravity force, arediscussed in Sec.4.
The equations governing fluid in absence of the first, second viscosity and thermal conductionmanifest conservation of momentum, energy and mass. They are generally nonlinear. We startfrom the linearized differential equations in terms of variations of pressure and density, p ′ and ρ ′ from hydrodynamically stable stationary functions p , ρ , which are not longer constants: ∂ −→ V∂t = − −→∇ p ′ ρ − −→ g ,∂p ′ ∂t = −−→ V · (cid:16) −→∇ p (cid:17) − γρ (cid:16) −→∇ · −→ V (cid:17) , (1) ∂ρ ′ ∂t = −−→ V · (cid:16) −→∇ ρ (cid:17) − ρ (cid:16) −→∇ · −→ V (cid:17) . The mean flow is absent, so that its velocity equals zero, −→ V ( x, y, z ) ≡ ~
0. The external forcewill be described by the gravity acceleration −→ g = (0 , , g ), though it may involve readily othermass forces including non-inertial ones. The flow of an ideal gas is considered, which internalenergy e in terms of pressure and density takes the form e = p ( γ − ρ , (2)where γ = C p /C v denotes the specific heats ratio. Eqs (1) describe gas motion of infinitelysmall amplitude. The background pressure and density follow from the zero order stationaryequality, dp ( z ) dz = − gρ ( z ) . (3)The background quantities supporting the equilibrium distribution of temperature T ( z ), takethe form ρ ( z ) = ρ (0) H (0) H ( z ) exp (cid:18) − Z z dz ′ H ( z ′ ) (cid:19) , H ( z ) = T ( z )( C p − C v ) g . (4)3t is well-established, that distribution of the background temperature is equilibrium if theparameter of static stability ν ( z ) is positive [4], ν ( z ) = γ − γ dH ( z ) dz > . (5)For some reason, it is convenient to introduce the quantity ϕ ′ instead of perturbation in density, ϕ ′ = p ′ − γ pρ ρ ′ . (6)The integral ε = 12 Z (cid:18) ρ −→ V + p ′ γp + ϕ ′ γν ( z ) p (cid:19) dv (7)is invariant, where v = {−∞ < x, y < ∞ , ≤ z ≤ h } , (8)and h may be infinity. It readily follows from Eqs (1)–(7), that ∂ε∂t = − Z −→∇ · ( p ′ −→ V ) dv = − I σ ( v ) p ′ −→ V d~σ = 0 , (9)where σ is a surface circumscriptive the volume v . The invariance of ε manifests conservationof the total energy of gas, which includes kinetic, barotropic and thermal parts. For ε to beinvariant, there is a certain freedom to establish the boundary conditions at z = 0 and z = h : V z ( z = 0) = V z ( z = h ) = 0, p ′ is any smooth function (impermeability condition across theboundaries) , or, for example, V z ( z = 0) = 0 , p ′ ( z = h ) = 0. Introducing the new set of variables, P = p ′ · exp (cid:18)Z z dz ′ H ( z ′ ) (cid:19) , Φ = ϕ ′ · exp (cid:18)Z z dz ′ H ( z ′ ) (cid:19) , ~U = ~V · exp (cid:18) − Z z dz ′ H ( z ′ ) (cid:19) , (10)one may readily rearrange Eqs (1) into the following set: ∂U x ∂t = − ρ (0) ∂P∂x ,∂U y ∂t = − ρ (0) ∂P∂y , (11) ∂U z ∂t = 1 ρ (0) (cid:18) γ − γH (0) − η ( z ) ∂∂z (cid:19) P + Φ γH (0) ρ (0) ,∂P∂t = − γgH (0) ρ (0) (cid:18) ∂U x ∂x + ∂U y ∂y + ∂U z ∂z (cid:19) − gρ (0) γ − η ( z ) U z ,∂ Φ ∂t = − ν ( z ) η ( z ) gρ (0) U z , where η ( z ) = H ( z ) H (0) . Eqs (11) determine the spectral problem.4 .1 Case of the constant background temperature The analytical analysis of the dispersion relations and determined by them modes may beproceeded in the case of constant T and hence constant H . In this case, η = 1 and ν = γ − ∂∂t Ψ( ~r, t ) = L (cid:18) ∂∂x , ∂∂y , ∂∂z (cid:19) Ψ( ~r, t ) , (12)where Ψ = [ U x U y U z P Φ ] T , (13) ~r = ( x, y, z ) and L is the matrix operator depending on spacial partial derivatives.The condition of algebraic solvability of Eqs (12) may be established by the Fourier trans-formation using the basis functions exp( ik x x + ik y y + ik z z ), Ψ( ~r, t ) = R ∞−∞ exp( − iωt + ik x x + ik y y + ik z z ) ψ ( k x , k y , k z ) dk x dk y dk z + cc , Det || iω I − l ( k x , k y , k z ) || = 0 , (14)where I is the unit matrix, l represents the matrix operator L in the space of Fourier trans-forms. The dispersion equation (14) determines five roots of dispersion equation, or dispersionrelations, everyone responsible for an especial type of gas motion. There are four wave modes,denoted by indices 1 , , ,
4, and the entropy mode, marked by 0: ω = 0 , (15) ω , = ± r γgH vuut k x + k y + k z + 14 H + s(cid:18) k x + k y + k z + 14 H (cid:19) − γ − γ H ( k x + k y ) ,ω , = ± r γgH vuut k x + k y + k z + 14 H − s(cid:18) k x + k y + k z + 14 H (cid:19) − γ − γ H ( k x + k y ) . Vectors of perturbations Ψ n ( n = 0 , . . .
4) correspondent to eigenvalues iω n , form the Hilbertspace L ( v ) with the scalar product h Ψ n , Ψ m i = Z (cid:18) ρ −→ U n · −→ U m + P n P m γgHρ + Φ n Φ m γ ( γ − gHρ (cid:19) dv (16)and the invariant E = 12 Z (cid:18) ρ |−→ U | + P γgHρ + Φ γ ( γ − gHρ (cid:19) dv, (17)where −→ U , P and Φ represent a sum of all parts of the eigenvectors, −→ U = X n =0 −→ U n , P = X n =0 P n , Φ = X n =0 Φ n . (18)The set Ψ n form complete set of eigenvectors. That is true for the self-adjoint boundaryconditions. It may be readily established, that iL is symmetric in L ( v ) (0 ≤ n, m ≤ h iL Ψ n , Ψ m i − h Ψ m , iL Ψ n i = i I σ ( v ) (cid:16) P n −→ U m + P m −→ U n (cid:17) d~σ = 0 . (19)The most important physically condition of impermeability at the upper and lower boundaries, z = 0 and z = h , is self-adjoint: U z ( z = 0) = U z ( z = h ) = 0. It is follow from Eqs (16)–(18)that E may be decomposed in to specific energies, E = P n =0 E n .5 Decomposition of acoustic and entropy modes in one-dimensional flow. General case
In one dimension ( k x = k y ≡
0) and constant T , the dispersion relations (15) determine threemodes, or, in the other words, possible motions in the gas. Two of them are acoustic, describingdifferent direction of sound propagation, and the last one is stationary, or entropy mode. It isresponsible for stationary variations in pressure and density which result in stationary changein the gas temperature. In the absence of mass force, this mode would be isobaric. In thecase of variable T , the dispersion equation, valid over all wave-length range, can not longer bewritten on, but the modes may be determined by relations linking field perturbations in thiscase as well. The completeness of the set of eigenvectors allow to represent the total vector ofperturbations as a sum of acoustic and entropy vectors at any instant,Ψ( z, t ) = X n =1 Ψ n ( z, t ) = Ψ ( z, t ) + Ψ a ( z, t ) , (20)where index a denotes summary acoustic vector. Eqs (11) yield the relation connecting P ( z, t )and Φ( z, t ) in both acoustic branches, P a = η ( z ) ν ( z ) (cid:18) γ − η ( z ) + γH (0) ∂∂z (cid:19) Φ a , (21)and the link for the stationary entropy mode,Φ = (cid:18) − γ −
22 + γH (0) η ( z ) ∂∂z (cid:19) P . (22)The scalar product of eigenvectors is given by equality h Ψ n , Ψ m i = Z (cid:18) ρ −→ U n · −→ U m + P n P m p + Φ n Φ m ν ( z ) p (cid:19) dv. (23)The total vector of perturbations is a sum of orthogonal modesΨ( z, t ) = U z P Φ = Ψ ( z, t ) + Ψ ( z, t ) + Ψ ( z, t ) ≡ Ψ a ( z, t ) + Ψ ( z, t ) = (24) U a,z (cid:16) γ − η ( z ) + γH (0) ∂∂z (cid:17) η ( z ) ν ( z ) Φ a Φ a + P (cid:0) − γ − + γH (0) η ( z ) ∂∂z (cid:1) P . Excluding P from the system (24), one obtains the equation describing R ( z, t ) = η ( z ) ν ( z ) Φ a ( z, t ): (cid:18) H (0) dη ( z ) dz − H (0) η ( z ) ∂ ∂z (cid:19) R ( z, t ) = 2 η ( z ) γ (cid:18)
2Φ + (cid:18) γ − − γH (0) η ( z ) ∂∂z (cid:19) P (cid:19) . (25)6ts solution consists of that of the homogeneous equation and a partial solution, R ( z, t ) = R Z z R ( z ′ ) D ( z ′ , t ) dz ′ − R Z z R ( z ′ ) D ( z ′ , t ) dz ′ + C R + C R , (26)where R ( z ), R ( z ) are determined by R ( z ) = exp (cid:18) − Z z dz ′ H η ( z ′ ) (cid:19) , R ( z ) = R Z z dz ′ R ( z ′ ) , (27)and D ( z, t ) = − H (0) η ( z ) γ (cid:18)
2Φ + (cid:18) γ − − γH (0) η ( z ) ∂∂z (cid:19) P (cid:19) . (28) R ( z, t ) uniquely determines P a and stationary quantities Φ = Φ − Φ a and P in accordance toEqs (24). H on z In this case, η = 1 + αz , α is some non-zero constant, and ν = 1 − γ + γαH (0), and R , R take the following form R ( z ) = (cid:18)
11 + αz (cid:19) / αH (0) , R ( z ) = H (0) (cid:18)
11 + αz (cid:19) / αH (0) − αz ) /αH (0) αH (0) . (29)Eqs (25)–(29) allow to distinguish uniquely acoustic and entropy mode in any vector of totalperturbations. Some simple conclusions follow immediately from Eqs (24). In this case, Φ ≡ Φ = (cid:18) − γ −
22 + γH (0)(1 + αz ) ∂∂z (cid:19) P. (30)In illustrations, we take P ≡ P in the form of a Gaussian impulse (a) and its derivative (b),( a ) P = π exp( − ( z − z ) /β H (0) ) , ( b ) P = − π exp( − ( z − z ) /β H (0) )( z − z ) H (0) β , (31)where β, π denote the characteristic width of the impulse in units H and its amplitude. Itwould be superfluous to mention, that relations between field perturbations specifying everymode, Eqs (24), are valid at any instant. So, in this subsection and two subsections below,we do not determine the time which the samples of perturbations relate to. For definiteness, z = 3 H (0) , β = 0 .
3, and γ = 1 .
4. 7 Π - zH H L - - F Π P Π - zH H L - - - - F Π a bFig.1 Excess pressure and entropy in the entropy mode for different αH (0) ( − . , , . In accordance to Eqs (24), P ≡ P a = (cid:18) γ − αz ) + γH (0) ∂∂z (cid:19) αz − γ + γH (0) α Φ . (32)Φ ≡ Φ a is a Gaussian pulse and its derivative,( a ) Φ = π exp( − ( z − z ) /β H (0) ) , ( b ) Φ = − π exp( − ( z − z ) /β H (0) )( z − z ) H (0) β , (33)The values of z , β , and γ are the same as in the above subsection. F a Π - zH H L - - P a Π F a Π - zH H L - P a Π a bFig.2 Excess pressure and entropy in the sound for different αH (0) ( − . , , . The specific perturbations in pressure for the stationary mode is taken in both forms (a,b) as inEq.(31), and the correspondent entropy is given by Eq.(30). Perturbation in entropy for soundis Φ a = − Φ , and the excess pressure relates to Φ a by means of Eq.(32). Fig.3 shows excesspressure for acoustic and entropy modes in the cases (a) and (b) for different α .8 P Π - zH H L - - P a Π P Π - zH H L - - - P a Π a bFig.3 Case of zero total entropy. αH (0) takes values ( − . , , . The choice of reference quantity in sound (Φ a ) was made due to simplicity of relation linking P a and Φ a for both acoustic branches. Note that in the limit of homogeneous gas ( g → H → ∞ , but the product gH (0) remains constant), γgH (0) is the squared sound of velocityin the uniform ideal gas (about the limit look the numerical investigation [13]). In this case, ϕ ′ means the quantity proportional to perturbation in the entropy, ϕ ′ = ( γ − ρT s ′ . It isidentically zero in both acoustic branches and is not longer suitable to be a reference quantityin them. Instead of, perturbation in density or in temperature may be chosen. In order toconclude about velocity of the acoustic mode, the knowledge of relation linking it with Φ a is required. The relations differ in sign for different acoustic modes. That follows from theconservation system (11): U ,z ( z, t ) = K Φ ( z, t ), U ,z ( z, t ) = − K Φ ( z, t ), where K is someintegro-differential operator. It may be concluded from these equalities and Eqs (24), that ifΦ ( z, t ) = − Φ ( z, t ) and P ( z, t ) = − P ( z, t ), the total field is represented by the entropy modeand acoustic field with non-zero velocity U z ( z, t ) = U ,z ( z, t ) + U ,z ( z, t ) ≡ U ,z ( z, t ) ≡ U ,z ( z, t )(hence, in general, the non-zero kinetic energy). Zero total kinetic energy means inevitablyequal perturbations in pressure and entropy of both acoustic branches, P ( z, t ) = P ( z, t ),Φ ( z, t ) = Φ ( z, t ). The relations (24) are valid at any instant . In the other words, thoughperturbations in every quantity varies with time, links inside every mode remain independenton time.To make approximate evaluations of K , one may consider the waveforms which spectrumincludes mostly large wavenumbers k z , k z H (0) >> ? ]. That allows to expand acoustic eigen-values in the Taylor series and do obtain finally relations for every acoustic branch, propagatingover the isothermal gas: U ,z ( z, t ) = − gγ ρ ( γ − γgH (0)) / Z z Φ ( z ′ , t ) dz ′ , (34) U ,z ( z, t ) = gγ ρ ( γ − γgH (0)) / Z z Φ ( z ′ , t ) dz ′ . The simple conclusion from Eqs (34) is that if R z (Φ ( z ′ , t ) + Φ ( z ′ , t )) dz ′ , the kinetic energy atthis instant is zero. 9he relations linking U z with P and Φ in a fluid affected by a mass force, are integro-differential. The exact links of excess pressure, density and velocity in unbounded volumes ofgas with constant H , are derived exactly with regard to one-dimensional flow by one of theauthors in [9, 10, 11]. They are valid for an ideal gas or any other fluid. For fluid different froman ideal gas, the equation of state, Eq.(2) should be corrected [10]. That corrects also dynamicequation governing an excess pressure, the second one from Eqs (1), and therefore, definitionof modes and even definition of P ,Φ and −→ U (Eqs (10)). It was shown in the study [10], that inthe case of constant H ≡ H (0), the relations (10) should be completed by the factor exp( ξz ),where ξ = − AgH + B H ( gH − B ) , (35)where A and B follow from the leading-order series of excess internal energy, e ′ = A p ′ ρ + B p ′ ρ . (36) ξ equals zero only in an ideal gas. Estimations for water at normal conditions result in ξ ≈ . / H . So that, even in a linear fluid flow over the background of constant temperature, theresults can not be generalized by common replacing γ by c ρ (0) /p (0), where c denotes the soundvelocity over the fluid without mass force, and p denotes the unperturbed internal pressure init. Illustrations of Sec.5 relate to αH (0) taking values − .
1, 0 or 0 .
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