Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

Williams, Paul D.

doi:10.1175/mwr-d-12-00303.1

Cited by 29 publications

(33 citation statements)

References 37 publications

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“…). For example, there have been recent developments in stabilizing the computational mode of the commonly used leapfrog time‐stepping scheme using the Robert–Asselin–Williams (RAW) filter (Williams, ; ; ). Specifically, the use of the RAW filter, instead of the Robert–Asselin filter, reduces biases in the location of the Gulf Stream (Young et al.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty and scale interactions in ocean ensembles: From seasonal forecasts to multidecadal climate predictions

Zanna

Brankart

Huber

et al. 2018

Quart J Royal Meteoro Soc

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The ocean plays an important role in the climate system on time-scales of weeks to centuries. Despite improvements in ocean models, dynamical processes involving multiscale interactions remain poorly represented, leading to errors in forecasts. We present recent advances in understanding, quantifying, and representing physical and numerical sources of uncertainty in novel regional and global ocean ensembles at different horizontal resolutions. At coarse resolution, uncertainty in 21st century projections of the upper overturning cell in the Atlantic is mostly a result of buoyancy fluxes, while the uncertainty in projections of the bottom cell is driven equally by both wind and buoyancy flux uncertainty. In addition, freshwater and heat fluxes are the largest contributors to Atlantic Ocean heat content regional projections and their uncertainties, mostly as a result of uncertain ocean circulation projections. At both coarse and eddy-permitting resolutions, unresolved stochastic temperature and salinity fluctuations can lead to significant changes in large-scale density across the Gulf Stream front, therefore leading to major changes in large-scale transport. These perturbations can have an impact on the ensemble spread on monthly time-scales and subsequently interact nonlinearly with the dynamics of the flow, generating chaotic variability on multiannual time-scales. In the Gulf Stream region, the ratio of chaotic variability to atmospheric-forced variability in meridional heat transport is larger than 50% on time-scales shorter than 2 years, while between 40 and 48 • S the ratio exceeds 50% on on time-scales up to 28 years. Based on these simulations, we show that air-sea interaction and ocean subgrid eddies remain an important source of error for simulating and predicting ocean circulation, sea level, and heat uptake on a range of spatial and temporal scales. We discuss how further refinement of these ensembles can help us assess the relative importance of oceanic versus atmospheric uncertainty in weather and climate.

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Section: Introductionmentioning

confidence: 99%

Uncertainty and scale interactions in ocean ensembles: From seasonal forecasts to multidecadal climate predictions

Zanna

Brankart

Huber

et al. 2018

Quart J Royal Meteoro Soc

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“…Following Williams (), let us now assume that in a computational code both x ( t ) and

are kept in memory. Then we can use a linear combination of them to calculate the tendency associated with the slow term, which we write as

.…”

Section: Theoretical Analysismentioning

confidence: 99%

“…Amezcua et al () have found that the RAW filter improves the skill of medium‐range weather forecasts compared with the Robert–Asselin filter. Many current models have subsequently adopted the RAW filter in place of the Robert–Asselin filter (see Williams, for a list).…”

Section: Introductionmentioning

confidence: 99%

“…Williams () identified two strategies for increasing the amplitude accuracy of the filtered leap‐frog scheme further. We recall that the RAW filter eliminates the first‐order amplitude errors associated with the Robert–Asselin filter and yields third‐order amplitude accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that the RAW filter eliminates the first‐order amplitude errors associated with the Robert–Asselin filter and yields third‐order amplitude accuracy. The two improvements proposed by Williams () are as follows. Firstly, leap‐frogging over a suitably weighted blend of the filtered and unfiltered tendencies was shown to eliminate the third‐order amplitude errors and yield fifth‐order amplitude accuracy.…”

Section: Introductionmentioning

confidence: 99%

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The composite‐tendency Robert–Asselin–Williams (RAW) filter in semi‐implicit integrations

Amezcua

Williams

2014

Quart J Royal Meteoro Soc

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Time discretization in weather and climate models introduces truncation errors that limit the accuracy of the simulations. Recent work has yielded a method for reducing the amplitude errors in leap-frog integrations from first-order to fifth-order. This improvement is achieved by replacing the Robert-Asselin filter with the Robert-Asselin-Williams (RAW) filter and using a linear combination of unfiltered and filtered states to compute the tendency term. The purpose of the present article is to apply the composite-tendency RAW-filtered leapfrog scheme to semi-implicit integrations. A theoretical analysis shows that the stability and accuracy are unaffected by the introduction of the implicitly treated mode. The scheme is tested in semi-implicit numerical integrations in both a simple nonlinear stiff system and a medium-complexity atmospheric general circulation model and yields substantial improvements in both cases. We conclude that the composite-tendency RAW-filtered leap-frog scheme is suitable for use in semi-implicit integrations.

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