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The Batchelor–Howells–Townsend spectrum: Three-dimensional case (2022)
Journal Article
Jolly, M., & Wirosoetisno, D. (2023). The Batchelor–Howells–Townsend spectrum: Three-dimensional case. Physica D: Nonlinear Phenomena, 445, Article 133615. https://doi.org/10.1016/j.physd.2022.133615

Given a velocity field u(x, t), we consider the evolution of a passive tracer governed by @t + u · ∇ = + g with time-independent source g(x). When u is small in some sense, Batchelor, Howells and Townsend (1959, J. Fluid Mech. 5:134; henceforth BHT59... Read More about The Batchelor–Howells–Townsend spectrum: Three-dimensional case.

Tracer turbulence: the Batchelor--Howells--Townsend spectrum revisited (2020)
Journal Article
Jolly, M., & Wirosoetisno, D. (2020). Tracer turbulence: the Batchelor--Howells--Townsend spectrum revisited. Journal of Mathematical Fluid Mechanics, 22(2), Article 18. https://doi.org/10.1007/s00021-019-0478-6

Given a velocity field u(x, t), we consider the evolution of a passive tracer θ governed by ∂tθ + u · ∇θ = Δθ + g with time-independent source g(x). When u is small in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted t... Read More about Tracer turbulence: the Batchelor--Howells--Townsend spectrum revisited.

Navier–Stokes equations on the β-plane: Determining modes and nodes (2018)
Journal Article
Miyajima, N., & Wirosoetisno, D. (2019). Navier–Stokes equations on the β-plane: Determining modes and nodes. Physica D: Nonlinear Phenomena, 386-387, 31-37. https://doi.org/10.1016/j.physd.2018.08.005

We revisit the 2d Navier–Stokes equations on the periodic β-plane, with the Coriolis parameter varying asβy, and obtain bounds on the number of determining modes and nodes of the flow. The number of modesand nodes scale as c G1/20+ c′(M/β)1/2and c G2... Read More about Navier–Stokes equations on the β-plane: Determining modes and nodes.

Energy spectra and passive tracer cascades in turbulent flows (2018)
Journal Article
Jolly, M., & Wirosoetisno, D. (2018). Energy spectra and passive tracer cascades in turbulent flows. Journal of Mathematical Physics, 59(7), Article 073104. https://doi.org/10.1063/1.5046773

We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tra... Read More about Energy spectra and passive tracer cascades in turbulent flows.

Averaging method applied to the three-dimensional primitive equations (2016)
Journal Article
Petcu, M., Temam, R., & Wirosoetisno, D. (2016). Averaging method applied to the three-dimensional primitive equations. Discrete and Continuous Dynamical Systems - Series A, 36(10), 5681-5707. https://doi.org/10.3934/dcds.2016049

In this article we study the small Rossby number asymptotics for the three-dimensional primitive equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-calle... Read More about Averaging method applied to the three-dimensional primitive equations.

Timestepping schemes for the 3d Navier-Stokes equations (2015)
Journal Article
Hong, Y., & Wirosoetisno, D. (2015). Timestepping schemes for the 3d Navier-Stokes equations. Applied Numerical Mathematics, 96, 153-164. https://doi.org/10.1016/j.apnum.2015.05.006

It is well known that the (exact) solutions of the 3d Navier–Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity. They also remain bounded for a finite time for arbitrary i... Read More about Timestepping schemes for the 3d Navier-Stokes equations.

Navier-Stokes equations on a rapidly rotating sphere (2015)
Journal Article
Wirosoetisno, D. (2015). Navier-Stokes equations on a rapidly rotating sphere. Discrete and Continuous Dynamical Systems - Series B, 20(4), 1251-1259. https://doi.org/10.3934/dcdsb.2015.20.1251

We extend our earlier β-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity 1/ϵ becomes zonal i... Read More about Navier-Stokes equations on a rapidly rotating sphere.

Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics (2014)
Journal Article
Tone, F., Wang, X., & Wirosoetisno, D. (2014). Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics. Numerische Mathematik, 130(3), 541-566. https://doi.org/10.1007/s00211-014-0670-9

We consider a two-dimensional model of double-diffusive convection and its time discretisation using a second-order scheme (based on backward differentiation formula for the time derivative) which treats the non-linear term explicitly. Uniform bounds... Read More about Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics.

Some spectral applications of McMullen's Hausdorff dimension algorithm (2012)
Journal Article
Gittins, K., Peyerimhoff, N., Stoiciu, M., & Wirosoetisno, D. (2012). Some spectral applications of McMullen's Hausdorff dimension algorithm. Conformal Geometry and Dynamics, 16, 184-203. https://doi.org/10.1090/s1088-4173-2012-00244-5

Using McMullen's Hausdorff dimension algorithm, we study numerically the dimension of the limit set of groups generated by reflections along three geodesics on the hyperbolic plane. Varying these geodesics, we found four minima in the two-dimensional... Read More about Some spectral applications of McMullen's Hausdorff dimension algorithm.

Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations (2012)
Journal Article
Gottlieb, S., Tone, F., Wang, C., Wang, X., & Wirosoetisno, D. (2012). Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations. SIAM Journal on Numerical Analysis, 50(1), 126-150. https://doi.org/10.1137/110834901

This paper considers the long-time stability property of a popular semi-implicit scheme for the two-dimensional incompressible Navier–Stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicit... Read More about Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations.

Slow manifolds and invariant sets of the primitive equations (2011)
Journal Article
Wirosoetisno, D., & Temam, R. (2011). Slow manifolds and invariant sets of the primitive equations. Journal of the Atmospheric Sciences, 68(3), 675-682. https://doi.org/10.1175/2010jas3650.1

The authors review, in a geophysical setting, several recent mathematical results on the forced–dissipative hydrostatic primitive equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence an... Read More about Slow manifolds and invariant sets of the primitive equations.

Exponential approximations for the primitive equations of the ocean (2007)
Journal Article
Temam, R. M., & Wirosoetisno, D. (2007). Exponential approximations for the primitive equations of the ocean. Discrete and Continuous Dynamical Systems - Series B, 7(2), 425-440. https://doi.org/10.3934/dcdsb.2007.7.425

We show that in the limit of small Rossby number \varepsilon, the primitive equations of the ocean (OPEs) can be approximated by "higher-order quasi-geostrophic equations'' up to an exponential accuracy in \varepsilon. This approximation assumes well... Read More about Exponential approximations for the primitive equations of the ocean.

On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations (2006)
Journal Article
Tone, F., & Wirosoetisno, D. (2006). On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations. SIAM Journal on Numerical Analysis, 44(1), 29-40. https://doi.org/10.1137/040618527

In this paper we study the stability for all positive time of the fully implicit Euler scheme for the two-dimensional Navier--Stokes equations. More precisely, we consider the time discretization scheme and with the aid of the discrete Gronwall lemma... Read More about On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations.

Persistence of steady flows of a two-dimensional perfect fluid in deformed domains (2005)
Journal Article
Wirosoetisno, D., & Vanneste, J. (2005). Persistence of steady flows of a two-dimensional perfect fluid in deformed domains. Nonlinearity, 18(6), 2657-2680. https://doi.org/10.1088/0951-7715/18/6/013

The robustness of steady solutions of the Euler equations for two-dimensional, incompressible and inviscid fluids is examined by studying their persistence for small deformations of the fluid-domain boundary. Starting with a given steady flow in a do... Read More about Persistence of steady flows of a two-dimensional perfect fluid in deformed domains.

Sobolev and Gevrey regularity results for the primitive equations in three space dimensions (2005)
Journal Article
Petcu, M., & Wirosoetisno, D. (2005). Sobolev and Gevrey regularity results for the primitive equations in three space dimensions. Applicable Analysis, 84(8), 769-788. https://doi.org/10.1080/00036810500130745

The aim of this article is to present a qualitative study of the Primitive Equations in a three-dimensional domain, with periodical boundary conditions. We start by recalling some already existing results regarding the existence locally in time of we... Read More about Sobolev and Gevrey regularity results for the primitive equations in three space dimensions.