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A common way to introduce physical processes into numerical models of the atmosphere is to call the parameterization at every grid point. This can lead to considerable errors. A simple 1-D example is proposed to illustrate that when a physical process occurs at one grid point only, a considerable sampling error may occur, with the result that only a fraction of the true impact of this process is seen. The interface to the physical parameterization in numerical weather prediction model using a third-order 1-D spectral element method (SEM3) model is investigated by homogeneous advection. In SEM3, the grid points, called principal nodes, are at boundaries of computational intervals and two more collocation points in the interior of each cell. This article argues that it is sufficient to do the physical parameterization for principal nodes only that creating the interior grid-point values of physics schemes by linear interpolation. This is called the spline interface method. A simple condensation model of water is taken as an example. Compared to the standard paramaterization, which computes the physical processes at every grid point, the spline interface method is more accurate and has a potential to save computer time. It turns out that the standard method creates a noisy wave which can easily be filtered by hyperviscosity. In the spline interface to the condensation physics, the condensation is done at every third grid point only. Third-order spline methods are used to represent the condensation at other points. The method using a smaller grid to compute condensation represented the condensation process more accurately and produced less of the computational noise. This version could be run without hyperviscosity, as no significant computational noise mode was generated by condensation. By doing physical processes only at every third grid point computer time may be saved.

To represent the advection process with reasonable accuracy fields of sufficient smoothness are necessary. In virtually all realistic models of the atmosphere, the physical processes are called at all grid points (Mesinger et al.

The parameterized physical processes can generate rough fields, as in many current models, the parameters determining the physical processes may be chosen in an unsmooth way. In this way, the physical parameterizations create fields which the advection processes cannot handle. Therefore, it makes sense to compute the paramaterization on a reduced grid, for example only for every third grid point.

In order to explore this idea we use a one-dimensional (1-D) model of homogeneous advection of the density field _{i}

The Legendre-Gauss-Lobatto grid _{i}

The test equation is homogeneous advection:
_{0} is 1. For the approximation of

The aim of this article is to explore the performance of physical processes on a reduced grid for physics, consisting of every third point of the full grid. The physical process chosen as a test for the reduced grid physics calculation is condensation. A very simple condensation scheme for the 1-D model will be defined in

As pointed out in the introduction, it is questionable that the physics is conveniently called at every grid point, or that is reasonable to allow the parameters determining the physical parameterization to change discontinuously from one grid point to the next, as is common practice in realistic models (Steppeler et al.

When interpreting _{max}

Condensation after one time step for

When

To define this interface, the field is be partitioned into a piecewise linear part

The linear part of

The high-order part of

The test used here is constructed to investigate the performance of physical parameterization schemes for small-scale atmospheric structures combined with a physical parameterization which also acts on a small scale. The test will be performed on a grid with 600 points (200 grid intervals) with

Similar initial values were used by Steppeler and Klemp (_{i}

To appreciate the information on time stepping, it may be useful to know that RK4 with centered difference in space implies a CFL condition of 2.8 and this is reduced by a factor 0.7 when centered differences are replaced by standard fourth-order spatial differencing (Ullrich

_{300}. The correct solution should show a maximum of 2. The solution in

Advection without condensation. The solid line indicates

Full-grid condensation without hyperviscosity: the condensation is applied every grid point. Dashed lines indicate the

The experiments are continued without hyperviscosity. Spline condensation as described above computes condensation only at every third grid point and uses splines to describe the effect of condensation at the other points. The result is shown in

Condensation at every third grid point and use of spline method for the interior points. The solid line indicates

The experiment (

Condensation at every third grid point with spatial centered difference. The solid line indicates

While the simple condensation in our play model is computationally not expensive, realistic models may have expensive condensation schemes, involving five kinds of water phases. The interface can also be applied to other physical processes, such as Ritter and Geleyn (

For the third-order L-Galerkin spectral element method SEM3, two interfaces for condensation physics were tested. For the physical parameterization, a simple condensation method together with homogeneous advection was used. A rather unsmooth physics scheme was used, where condensation occurs at one grid point only. The standard version, when condensation is done for every grid point, encountered a large sampling error. The spline condensation method would do the condensation at principal nodes only but extending it to the linear spline part of the field. It turned out that using the full-grid method created a 2

One advantage of the spline condensation method is the reduced need for fourth-order diffusion. Also, this method saves computer time, as for our 1-D case the condensation is done on only

The authors thank the city of Erquy, Brittany, France for the support of the visit by providing office space.

No authors report any potential conflicts of interest.