Temporal and Spatial Resolution

Now that you've decided on a three-dimensional model, and can choose the domain size that is appropriate for the problem at hand, the next set of choices you need to make have to do with the resolution in space and time.

The fundamental problem is that the equations that govern the motion of the atmosphere and ocean are continuous. In order to solve them on a computer, we need to discretize the domain, that is, create a grid system of points on which to solve the equations.

The schematic below shows a typical model grid system.

The variables of interest, the temperature, salinity, and velocity components, are solved at the grid points, denoted by green dots in the schematic. The distance between two grid points at which say temperature is solved is denoted by x. Likewise, the distance between two grid points in the y direction (usually latitude), is denoted by y. The distance between two depth or pressure grid points in the vertical, is denoted by z. The distance between grid points can be variable. For example, finer vertical resolution is normally required near the ocean surface, and finer horizontal resolution near coasts, where the variability is greater. Therefore, many models have smaller grid spacing in the vertical near the surface than say below the thermocline (region of greatest temperature change). Note that z is always less than x and y; there is no requirement for x to equal y, although often it does. Later, we will discuss several different ways the vertical grid can be represented.

How do you choose the resolution, that is, the x, y, and z values for a model? There are competing aspects to consider when deciding how fine a resolution to choose, that is, what the grid spacing and time step of the model should be.

1. The phenomena of interest need to be resolved.
   
   To resolve a wave, you need at least 4 points per wavelength, but more than that are usually used.
   
   Example: Suppose we have a rectangular domain for the North Atlantic ocean circulation that is 1500 km by 1500 km by 4000 m deep. What kind of resolution in x and y do we need to have? It depends on what phenomena we're trying to resolve. Let's suppose we need to be able to resolve the eddies that are formed and shed by the Gulf Stream. These eddies are characteristically 200 to 500 km in size, and persist for one to a few months. In order to resolve the motion of an eddy, the horizontal grid resolution would need to be at least 50 km, but 10 to 25 km is more typical.
2. Computer resources are limited. Until recently, it was impossible to even consider an eddy-resolving global ocean model. Now, with the advent of multi-processing large computers and incredible processor speed-up and capacity, it is possible to run fine-scale simulations for the global ocean. However, there are limitations if the predictions are needed for operational purposes; for example, a model that required 2 days to complete a 24-hour forecast would not be operationally useful.
3. The numerical scheme must be computationally stable. Finally, once the resolution in the 3 spatial dimensions has been set, the time step for the computation needs to be considered as well. In the numerical schemes that are used to solve the equations, the time step must be small enough for computational stability. One rule of thumb that applies is known as the Courant-Friedrich-Lewy, or "CFL", criterion. This is a stability condition that states that the speed the information is propagating at, c, times the time step, t must be less than the horizontal grid spacing, x. In other words, the wave (or advective process) must not move farther than 1 grid point per time step. This can be written mathematically as:
   c t < x.
   Using this relationship, we can obtain an upper limit on the value of the time step, t, which is
   t < x / c.
   
   
   "Explicit schemes allow one to advance the solution to the next time level, one spatial grid point at a time, and are quite simple to implement." (Kantha and Clayson, 2000). In an implicit time-stepping scheme, the solution at the next time level must be derived for all grid points simultaneously. These schemes are computationally more intensive, but are unconditionally stable, thus permitting larger time steps to be taken than would otherwise be required.

The activity you are about to do will illustrate the trade-offs among these three aspects. Your job is to set the time step, t, and the grid spacing, x, such that the problem is computationally stable, and the velocity is optimally resolved in time and space, without wasting computer time.

When you're ready, do the online activity.

As the previous activity demonstrated, much smaller time steps are needed for a given spatial resolution, when the advective, or wave, speed c is greater. In the ocean, 200 m/s is a typical speed for surface (also called external, or barotropic) gravity waves, while 2 m/s is more typical for internal, or baroclinic, gravity waves. Instead of solving the full set of primitive equations at the small time step needed for fast surface gravity waves, a method called mode-splitting, is commonly used to improve computational efficiency. Only the barotropic, or shallow water equations (discussed earlier), which are simpler than the full primitive equations, are solved using the short time step, while the more complicated equations for the baroclinic modes are solved using a slower time step.

There are other more subtle implications of the choice of model resolution involving computational truncation and round-off errors, but we will not go into those here.

Be aware that you cannot determine the temporal and spatial resolution of a model solely by looking at the graphical, or other, output. Often, the model results are subsampled in time and/or space for output, and occasionally they are interpolated (so the output has finer resolution than the model grid or time step).

Next, we'll look at various horizontal and vertical grid types.