# Given the well known effects of the inverse turbulence cascade for 2D flow, does it make sense to use grid resolutions that resolve turbulence?

As is well known, 2D simulations of turbulent flows in the ocean follow an inverse energy cascade, as opposed to a forward energy cascade observed in fully 3D flows. Because in 2D fluid filamentation is enhanced leading to strong density gradients that eventually become grid scale (where most of the mixing is owed to numerical mixing), the dissipation is underpredicted while mixing is overpredicted. As such, is there any value to running a 2D simulation with resolutions that attempt to resolve turbulence?

• Does your flow actually behave like 2D turbulence or are you saying using 2D turbulence (inverse cascade) is causing the error? – casey Oct 9 '14 at 10:20
• I am aware that there are flows that can behave like 2D turbulence (such as geostrophic turbulence). But I'm talking about in general, if we know that it is a true 3D flow, is there any point in using highly resolved 2D simulations that attempt to resolve the scales that are typically involved in turbulence? – Isopycnal Oscillation Oct 9 '14 at 18:04

Any numerical model solution is inherently constrained by the equations being solved. You are probably aware that the inverse turbulence energy cascade exists in predominantly 2D flows. In oceans and atmospheres, this happens at approximate scale of Rossby radius of deformation and larger. Whether or not a numerical model is able to represent a physical process (i.e. 3D turbulence) really depends on the terms in the model equations.

Take for example the simplest 2D model representative of ocean and atmosphere, shallow water equations (SWE, writing non-conservative form for brevity here so don't give me a hard time):

$$\dfrac{\partial \mathbf{v}}{\partial t} = -\mathbf{v}\cdot\nabla\mathbf{v} - f\mathbf{k}\times\mathbf{v} -g\nabla h$$

$$\dfrac{\partial h}{\partial t} = -\nabla(h\mathbf{v})$$

where $\mathbf{v}$ is the 2D velocity, $f$ is Coriolis frequency, $g$ is gravitational acceleration and $h$ is the fluid depth.

The linearized form of the SWE, i.e. $\mathbf{v}\cdot\nabla\mathbf{v}\approx0$ and $\nabla(h\mathbf{v})\approx H\nabla\cdot\mathbf{v}$, where $H$ is the mean fluid depth, has an analytical solution and is well known.

The non-linear term $\mathbf{v}\cdot\nabla\mathbf{v}$ allows for the fast growth of small perturbations, eventually resulting in eddy formation. In this model, the only dissipation is of numerical kind, and the only turbulence is of the 2D kind, supporting inverse energy cascade. In order to allow for 3D turbulence, a prognostic equation for vertical velocity $w$, including non-linear terms $\mathbf{v}\cdot\nabla w$ must be introduced to the model.

if we know that it is a true 3D flow, is there any point in using highly resolved 2D simulations that attempt to resolve the scales that are typically involved in turbulence?

If by this you mean, can I model 3D turbulence with a very-high resolution 2D model, the answer is no. 2D models are obtained by making an assumption that variations in the 3rd dimension are negligible compared to variations in the other 2 dimensions. It is this assumption that prohibits any 3D process to be part of the model solution. Sure, one can crank up horizontal resolution as much as they like, and the model will produce a solution, however that solution will always be inherently constrained by the initial assumptions made.

• I like how you stress the importance of vortex stretching. The reason why I asked was because I was thinking that in 2D numerical simulation, numerical dispersion and numerical diffusion concerns apart, there seems to be no difference between simulating a large scale or small scale flow (of course if one wants to know about a specific area, one could refine the mesh there, but that is a different thing). – Isopycnal Oscillation Oct 13 '14 at 8:27
• Question was also motivated because of many papers studying the 2D boundary layer numerically. It does not make much sense to me given that the turbulence values are all wrong. – Isopycnal Oscillation Oct 13 '14 at 8:34
• @IsopycnalOscillation Can you explain further what you mean by your second comment? Boundary layer models parameterize the effects of turbulence based on the state of the mean flow. Though they don't resolve 3D turbulence, they must be 3D models, because vertical mixing is typically parameterized as function of $\partial U/\partial z$, no? – milancurcic Oct 13 '14 at 19:03
• Yes I think that is correct, I'm just being pedantic. I just see no point in parameterizing the effects of turbulence, if for example, one is interested in quantities such as irreversible mixing, then the models are not really going to be that accurate. – Isopycnal Oscillation Oct 13 '14 at 19:37
• @IsopycnalOscillation there are different methods of parameterizing turbulence and various methods of resolving some of the turbulent scales (e.g. LES setup to resolve production scales but parameterize dissipation scales). Without knowing the details of your model it is hard to make a blanket statement about parameterization. – casey Oct 14 '14 at 13:04