Space-Time Regularized Navier

SPACE AND TIME REGULARIZED NAVIER–STOKES EQUATIONS

Nondissipative Space-Time Regularizations for the ICON Model

The regularized continuous problem
The time regularization
The space regularization
Numerical experiments
Selected bibliography

The regularized continuous problem

The traditional approach in numerical modeling is

define the continuous problem;
provide a numerical discretization for such problem.

In our case, however, an intermediate step is introduced:

define the continuous problem;
define a regularized version of the continuous problem;
discretize the regularized version of the continuous problem.

In a general sense, the regularization step is introduced to obtain a problem which is more suitable for the numerical discretization compared to the original continuous problem.

We are currently investigating two types of regularizations, aiming at two independent goals. The time regularization improves the efficiency of the numerical discretization by enlarging the stable time step, thus representing an alternative to the semi-implicit method, while the space regularization modifies the energy and enstrophy cascade toward small scales, thus representing an alternative to the usual (dissipative) turbulence closures.

Both the time and the space regularizations are nondissipative, since they do not enhance the viscosity of the system.

The time regularization

In the time regularization, a regularized pressure field is introduced in order to slow down the velocity of the fastest, acoustic waves. This results in a significant increase of the stable time step for an explicit time integration method, thus improving the efficiency of the overall numerical computation.

For the special case of the shallow water equations the regularized system reads

where v is the fluid velocity, f is the Coriolis parameter, g is the gravitational constant and h is the free surface elevation. The regularized free surface height is obtained from

where &tau is a constant characterizing the regularization and depending on the time step.

A more detailed description of the time regularized system can be found in^{[
8
9
10
11]}.

The space regularization

In the space regularization, a regularized velocity field is introduced in order to reduce the energy and enstrophy cascade toward small wave numbers. The resulting equation set has been found to be a promising starting point for the construction of ocean models in^{[
3
7
]}, where is has been shown that the use of the regularized equations yields result comparable with the unfiltered Navier–Stokes equations at twice the space resolution.

For the special case of the shallow water equations the regularized system reads

where &alpha is a constant which characterize the regularization and the regularized flow velocity is obtained as the solution of

A more detailed description of the space regularized system can be found in^{[
1
2
4
5
6
13]}.

Numerical experiments

Rossby–Haurwitz wave

We compare here the solutions obtained with the ICON-SW code for test case 6 of^[12] solving the plane shallow water system, the time regularized system and the space regularized system.

Four plots representing contour lines of the free surface elevation at day 12 for the analytic solution of the barotropic vorticity equation, the reference unfiltered shallow water solution, the time regularized solution and the space regularized solution.
Both the time and space regularized computations use a time step which is 2.67 times larger than the reference unfiltered computation. It can be seen that, on the one hand, the time regularization does not change the velocity of the Rossby–Haurwitz wave, while the space regularization does, as observed in^[13]. The space regularization also delays the break-down of the wave pattern.

Decaying turbulence experiment

To assess the impact of the space regularization on the energy cascade we consider a shallow water decaying turbulence experiment, where the time evolution of an initial random flow is simulated without external forcing.

Energy dissipation (E(t)-E(0))/E(0) during the first 30 days of simulation and energy spectra at the final time of 30 days for a decaying turbulence experiment for the unfiltered shallow water equations with viscosity 891 m²/s (blue) and 446 m²/s (black) and for the space regularized shallow water equations with viscosity 446 m²/s and &alpha=20 km (red).
It can be observed that:

the space regularization does not enhance the overall dissipativity of the model;
the spectrum of the regularized system follows the unfiltered spectrum corresponding to the same viscosity at low wave numbers and the spectrum corresponding to twice the viscosity at high wave numbers.

Selected bibliography

^[1] S. Chen, D. Holm, L. Margolin, and R. Zhang, Direct numerical simulations of the Navier–Stokes alpha model, Physica D: Nonlinear Phenomena, 133 (1999), pp. 66–83.

^[2] C. Foias, D. Holm, and E. Titi, The Navier–Stokes-alpha model of fluid turbulence, Physica D: Nonlinear Phenomena, 152-153 (2001), pp. 505–519.

^[3] M. Hecht, D. Holm, M. Petersen, and B. Wingate, Implementation of the LANS-&alpha turbulence model in a primitive equation ocean model, J. Comp. Phys., 227 (2008), pp. 5691– 5716.

^[4] D. Holm and B. Wingate, Baroclinic instabilities of the two-layer quasigeostrophic alpha model, J. Phys. Ocean., 35 (2005), pp. 1287–1296.

^[5] E. Lunasin, S. Kurien, M. Taylor, and E. Titi, A study of the Navier–Stokes-&alpha model for two-dimensional turbulence, J. Turb., 8 (2007), pp. N30–.

^[6] K. Mohseni, B. Kosovic, S. Shkoller, and J. Marsden, Numerical simulations of the Lagrangian averaged Navier–Stokes equations for homogeneous isotropic turbulence, Phys. Fluids, 15 (2003), pp. 524–544.

^[7] M. Petersen, M. Hecht, and B. Wingate, Efficient form of the LANS-&alpha turbulence model in a primitive-equation ocean model, J. Comp. Phys., 227 (2008), pp. 5717–5735.

^[8] S. Reich, Linearly implicit time stepping methods for numerical weather prediction, BIT Numerical Mathematics, 46 (2006), pp. 607–616.

^[9] S. Reich, N. Wood, and A. Staniforth, Semi-implicit methods, nonlinear balance, and regularized equations, Atmos. Sci. Lett., 8 (2007), pp. 1–6.

^[10] A. Staniforth and N. Wood, Analysis of the response to orographic forcing of a timestaggered semi-Lagrangian discretization of the rotating shallow-water equations, Q. J. R. Meteorol. Soc., 132 (2006), pp. 3117–3126.

^[11] A. Staniforth, N. Wood, and S. Reich, A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations, Q. J. R. Meteorol. Soc., 132 (2006), pp. 3107–3116.

^[12] D. Williamson, J. Drake, J. Hack, R. Jakob, and P. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comp. Phys., 102 (1992), pp. 211–224.

^[13] B. Wingate, The maximum allowable time step for the shallow water model and its relation to time-implicit differencing, Mon. Wea. Rev., 132 (2004), pp. 2719–2731.

(mre, 27.10.2007)