Pis'ma v ZhETF, vol. 97, iss. 3, pp. 145-149 © 2013 February 10
Direct numerical experiment on measuring of dispersion relation for gravity waves in the presence of condensate
A. O. Korotkevich1
Department of Mathematics and Statistics, University of New Mexico, MSC01 1115, 1 University of New Mexico, Albuquerque, NM 87131-0001, USA
Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia
Submitted 3 December 2012 Resubmitted 11 December 2012
During previous numerical experiments on isotropic turbulence of surface gravity waves we observed formation of the long wave background (condensate). It was shown (Korotkevich, Phys. Rev. Lett. 101(7), 074504 (2008)), that presence of the condensate changes a spectrum of direct cascade, corresponding to the flux of energy to the small scales from pumping region (large scales). Recent experiments show that the inverse cascade spectrum is also affected by the condensate. In this case mechanism proposed as a cause for the change of direct cascade spectrum cannot work. But inverse cascade is directly influenced by the linear dispersion relation for waves, as a result direct measurement of the dispersion relation in the presence of condensate is necessary. We performed the measurement of this dispersion relation from the direct numerical experiment. The results demonstrate that in the region of inverse cascade influence of the condensate cannot be neglected.
DOI: 10.7868/S0370274X13030028
Theory of wave or weak turbulence [1] applied to gravity waves on the surface of the fluid is the base for all current wave forecasting models, which are crucial for ocean cargo communications and oil and gas sea platforms operations. This is why it's verification is important and urgent problem. Numerous attempts to get a spectrum of the direct cascade of energy from ocean and sea observations give results which confirm the wave turbulence theory [2, 3]. At the same time all these experiments were working with wind generated waves, which means broad spectrum pumping. As a result it is hard to understand where we have so called "inertial interval" (region of scale where we have only nonlinear interaction of waves, while pumping and damping influences are negligible). Narrow frequency pumping can be realized in experimental wave tanks and flumes. But this state of the art experiments were producing very strange and contradictory results [4, 5], like dependence of the spectral slope on the amplitude of pumping. The direct numerical simulation of the primordial dynamical equations looks like a natural remedy for this problem. It provides us with all possible information about the system, but for the cost of enormous computational complexity.
One of the first attempts was pioneering work [6], which soon was followed by [7-9]. During the last
e-mail: alexkor@math.unm.edu
decade, the author together with colleagues were able to find answers at least to some of the open questions using direct numerical simulation of gravity waves. It was shown, that on a discrete grid of wavenumbers (common situation for both pseudo spectral numerical codes in a periodic domain and experimental wave tanks which are usually rectangular basins of finite size) the meso-scopic turbulence can take place [10, 11]. In the recent works [12, 13] the author demonstrated that formation of the inverse cascade, corresponding to the flux of wave action (analog of number of waves), inevitably leads to the formation of the strong long wave background, which we call condensate (due to similarity with Bose-Einstein condensation in condensed matter physics). It was shown that presence of the condensate changes the slope of the direct cascade spectrum. In these recent experiments, in spite of the short inertial interval for the inverse cascade, the slope significantly different from the predicted by the wave turbulence theory was observed. Recent reports [14] with long enough inertial interval show that the slope indeed differs from the theoretically predicted one.
In the present Letter we report results of direct measurement of the dispersion relation of the surface gravity waves in the presence of condensate. As it will be demonstrated later, the slope of the spectrum of the inverse cascade directly depends on the power of the dispersion relation. Although we were not able to determine the
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change of this power, we demonstrate that in the region of the inverse cascade the dispersion relation is strongly affected by the nonlinear interaction with the condensate. This distortion cannot be neglected and might be considered as a possible cause of the change of the slope of the inverse cascade spectrum.
As in many previous works [15, 7, 9, 12, 13] we shall consider isotropic turbulence (no direct dependence on angle) of gravity waves on the 2D surface of the 3D fluid. The isotropic turbulence is a wonderful sandbox for the turbulence simulation. In this case we can use angle averaging of the resulting spectra in order to decreases it natural peakedness of harmonics, instead of averaging over different realizations which is unrealistic in this case, due to a long time of even one simulation.
Here and further we shall follow notations from [12]. We consider a potential flow of ideal incompressible fluid. System is described in terms of weakly nonlinear equations [1] for surface elevation r(r,t) and velocity potential at the surface ^(r,t) (r = (x, y))
n = k^ — — k[nk'^>] + k(nk[nk'^]) +
r<P = - gv - T2
(V^)2 - (k^)2 - [k^\k[nk^\- (1)
— [r,k^]A^ + F-l[lk ^k] + Pr.
Here dot means time-derivative, A - Laplace operator, it is a linear integral operator (jt = i/—A^j, F is an inverse Fourier transform, jk is a dissipation rate (according to recent work [16] it has to be included in both equations), which corresponds to viscosity on small scales and, if needed, "artificial" damping on large scales. Pr is the driving term which simulates pumping on large scales (for example, due to wind). In the k-space supports of Yk and Pk are separated by the inertial interval, where the Kolmogorov-type solution can be recognized. These equations were derived as results of Hamiltonian expansion in terms of k,. From physical point of view k-operator is close to derivative, so we expand in powers of slope of the surface. In most of experimental observations average slope of the open sea surface n is of the order of 0.1, so such expansion is very reasonable. Additional details can be found in [7,17-19].
In the case of statistical description of the wave field, Hasselmann kinetic equation [20] for the distribution of the wave action n(k,t) ^ (|ak(t)|2) is used. Here
ak
2k^ + i
k
2uk
^k
(2)
are complex normal variables. For gravity waves uk = = \fgk. In this variables, if we have a linear wave with
wavenumber k, it will correspond to the only excited harmonics ak. In other words, representation in terms of these normal variables means representation in terms of elementary excitations in the system (linear waves). In reality we should use pair correlator for variables after canonical transformation which eliminates nonreso-nant terms in the Hamiltonian [1, 17], but in the case of gravity waves of average steepness vW> - 0.1 their relative difference is of the order of few percents. Thus, we neglect this difference and will be working with correlation function given above.
Numerical simulation. We simulated primordial dynamical equations (1) in a periodic spatial domain 2n x 2n. Main part of the simulations was performed on a grid consisting of 1024 x 1024 knots. Also we performed long time simulation on the grid 256 x 256. The used numerical code [21] was verified in [18, 7, 9, 10, 22, 23, 12, 13]. Gravity acceleration was g = 1. Pseudo-viscous damping coefficient had the following form
Yk
0,k < kd,
-Yo(k - kd)2, k > kd,
(3)
where kd = 256 and y0,i024 = 2.7 ■ for the grid 1024 x 1024 and kd = 64 and y0,256 = 2.4 ■ 102 for the smaller grid 256 x 256. Pumping was an isotropic driving force narrow in wavenumbers space with random phase:
Pk = fke
iRk(t)
4F0
(k - kpl)(kp2 - k)
, fk
(kp2 - kpi)2 0 - if k < kpi or k > kp2 ;
(4)
here, kp1 = 28, kp2 = 32, and Fo = 1.5 ■ 10~5; Rk(t) was uniformly distributed random number in the interval (0,2n] for each k and t. Initial condition was low amplitude noise in all harmonics. Time steps were Atio24 = 6.7 ■ 10-4 and At256 = 5.0 ■ 10-3. We used Fourier series in the following form:
2n 2n
nk = F[nr] =
1
(2n)
nreikrd2v,
o o
Nx/2 Ny/2
nr = F-1[nk]= E E nke-ikr,
-Nx/2 -Ny/2
here Nx, Ny are numbers of Fourier modes in x and y directions.
As results of simulation we observed [12, 13] formation of both direct and inverse cascades (Fig. 1, solid line). Average steepness was equal to ^/WW) = °-14-What is important, development of inverse cascade spectrum was arrested by discreteness of wavenumbers grid
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grid 1024x1024
Fig. 1. Spectra (\ak|2). With condensate on the 1024x1024 grid (solid); on the 256 x 256 grid with more developed condensate (long dashes)
in agreement with [18, 10, 24, 25]. Then large scale condensate started to form. The mechanism of condensate formation is the following. We have a flux of wave action (number of waves) from the pumping to the large scale region. This flux is due to nonlinear resonant interaction of waves. In [10] it was shown, that on a discrete grid of wavenumbers, which is typical for both finite experimental wave tanks and computer simulations, resonance conditions are never fulfilled exactly. What makes it possible for them to exist is the finite width of the resonant curve due to nonlinear frequency shift. As a result, this thick resonant curve covers some knots of the wavenumbers grid [18]. The nonlinear frequency shift is proportional to the matrix element (coupling coefficient) of interaction of waves, which is homogeneous function of the order 3 (with
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