1. The RENUMERATE project
Alex Megann1 and Maria Luneva2
(1) National Oceanography Centre, Southampton, UK
(2) National Oceanography Centre, Liverpool, UK
CMEMS Mid-Term meeting, 18 March 2019

2. RENUMERATE: Reducing numerical mixing resulting from applying tides explicitly in a global ocean model
Overall objectives and team presentation
Adding tides to ocean models
Model description
The tidally forced experiment
Summary of mixing analysis
Numerical mixing in fixed-coordinate models
The z~ coordinate
Preliminary results
Further experiments
Issues encountered
Projected impacts on CMEMS
Communications

3. Overall objectives of RENUMERATE project
Objective 1: To add explicit tidal forcing to the GO8.0 global ¼° NEMO configuration and evaluate the effects on water mass characteristics and mixing.
Objective 2: To add the z-tilde semi-lagrangian coordinate to this configuration and evaluate the effects on water mass characteristics and mixing, in particular to assess its effect in ameliorating numerical mixing from tidal and other high frequency motions.
Objective 3 To investigate the effects of tides on large scale ocean circulation, including global stratification, the overturning and gyre circulations; shelf slope exchange, seasonal mixed layers, deep and bottom waters, and polar sea ice.

4. The RENUMERATE team Alex Megann Maria Luneva Interests: Interests:
Model development, numerical mixing, vertical coordinate issues, ocean heat uptake
Role in project:
Overall PI; carry out model runs and evaluate them; lead on project deliverables
Maria Luneva
Interests:
High latitude oceans, ocean mixing schemes, tides, vertical coordinate issues
Role in project:
Test tidal configurations and diagnostics; advice on tides and mixing; contribute to reports and papers

5. Benefits of simulating tides in a global ocean model
Tidal motions close to ridges and rough bathymetry cause strong mixing in the ocean; standard parameterisations of tidal mixing can only be an approximation.
Tidal flows are dominant source of mixing on shelves.
Tides are responsible for a large part of transport of nutrients on and off shelves.

6. Potential downsides to tidal forcing
Large barotropic tidal velocities will stress stability of model numerics, especially in shallow shelf regions and over slopes.
Most global models, with horizontal resolutions of km and with vertical levels, cannot realistically simulate the internal tide, nor the explicit mixing from the latter.
Vertical motions from tides have potential to cause significant levels of numerical mixing in a depth-coordinate ocean model

7. The GO8p0.1 model configuration
Testbed for development of UK ocean configuration beyond CMIP6
Based on GO6/GO7 (used in GC3.1 and UKESM1 CMIP6 models)
Code base updated to NEMO v4, and CICE sea ice model replaced with new NEMO ice model SI3
Workhorse for development is on eORCA025 ¼° global grid and has 75 vertical levels
Default vertical coordinate is nonlinear free surface (z*): vertical levels are allowed to flex with the external mode

8. Adding tidal forcing to GO8p0.1
Tidal forcing was enabled in the model (M2, S2, N2, K1 and O1).
Default minimum depth is 9.5 metres: this caused failures in the model, primarily on Patagonian Shelf, but also elsewhere. Minimum depth was increased to 23 metres, and this resulted in a stable configuration.
Initial integration disabled the tidal mixing parameterisation, but in further experiments this was included, as simulated tides were deemed unlikely to contribute enough explicit mixing
Tidal configuration has been integrated for 30 years

9. First assessment of tidal motions
GO8p0 tidal experiment
GESLA analysis
RMS hourly surface elevation excursion from the mean (metres) over three months in tidally-forced simulation (left) and M2 amplitude in GESLA analysis (right)

10. Evolution of tidal signal
Large excursions (>2 metres) visible around NW Europe and Eastern Seaboard (esp. Bay of Fundy)
Three days of hourly surface elevation anomaly (m) in tidal model
NB: Movie removed by Mercator
for distribution (size issue)

11. Numerical mixing in fixed coordinate models
Vertical displacements of isopycnals (heave) due to waves and eddies --> cross-coordinate motion, causing diapycnal mixing.
This is a significant contribution to numerical mixing in global eORCA025 NEMO configurations (Megann, Ocean Modelling, 2018)
Time

12. Mixing diagnostics
We shall use the method of Lee et al (2002) and Megann (2018) to evaluate effective diffusivity keff, defined as follows:
where
and Y(Q,r) is the overturning streamfunction and V the volume beneath isopycnal r and north of latitude Q.
For each experiment, compare keff with:
explicit vertical diffusivity kexp ; and with
diagnosed diffusivity in control.

13. Assessing numerical mixing in tidal run
Ratio of effective diffusivity keff in integration with tidal forcing to keff in control
Vertical axis is potential density s2
Bold black dashed line is maximum surface density at given latitude
As expected, effective diffusivity is generally higher with tides than in control, particularly in intermediate water densities.

14. Tidally-forced model: summary
We have created a tidally forced implementation of the GO8p0.1 global ¼° NEMO model, and successfully integrated it to 30 years
Once minimum depth of shelves is increased to 23m, the tidal model is stable at the default timestep of 15 minutes.
Evaluation of RMS surface elevation anomaly confirms that barotropic tide is acceptably realistic
Preliminary estimate of effective diapycnal diffusivity suggests that level of numerical mixing is larger than that in simulation without tides

15. Tidally-forced model: next steps
Carry out harmonic analysis of surface elevation to compare with reanalyses
Carry out 3-D harmonic analysis to characterise internal tide
Complete watermass analysis
Evaluate effect of tides on on- and off-shelf transports

16. Potential solutions for numerical mixing
Less diffusive vertical advection schemes (e.g. vPPM)

17. Potential solutions for numerical mixing
Less diffusive vertical advection schemes (e.g. vPPM)
Suppress grid-scale vertical velocity noise in ¼° config by increasing viscosity (Megann and Storkey, JAMES, under review)

18. Potential solutions for numerical mixing
Less diffusive vertical advection schemes (e.g. vPPM)
Suppress grid-scale vertical velocity noise in ¼° config by increasing viscosity (Megann and Storkey, JAMES, under review)
Introduce elasticity in vertical coordinate so that it flexes with disturbances of the isopycnals on timescales less than a few days (z~). Leclair and Madec (2011) found this reduced numerical mixing by a factor of ~5 in an idealized model
We will discuss the latter approach here.

19. z~ at work: idealised configuration
Implementation in v3.6 with realistic bathymetry was unstable
Jerome has supplied modifications of z~ code in NEMO v4.0-beta
Vertical grid distortion associated with passage of large internal wave in NEMO with z~ scheme enabled (from Leclair and Madec, 2011)

20. First implementation of z~ in global model
Jerome’s code improvements were incorporated into the NEMO v4 GO8p0.1 configuration.
The first integration crashed in year 23 with sea ice thickness of over 90m in the Ross Sea. This was traced to a bug in the iceberg code, which was fixed by Pierre Mathiot at the Met Office.
A second integration, with the bug fixed, completed 30 years with no problem.
Cutoff time in initial configuration is 5 days

21. First evaluation of z~ in GO8
If z~ is working as a semi-lagrangian coordinate, it should move most short-timescale variance of vertical velocity into variance of interface depth (more like an isopycnic model!)
Compare RMS vertical velocity and interface depths in control and in z~ run

22. Reduction in advective vertical velocities
NB: Movie removed by Mercator
for distribution (size issue)
GO8p0.1 control
GO8p0.1 with z~
Hourly vertical velocity (m day-1) at 2,000m

23. Displacement of interface depths
NB: Movie removed by Mercator
for distribution (size issue)
GO8p0.1 control
GO8p0.1 with z~
Hourly interface depth anomaly of k=54 (approximately 2,000m)

24. Global temperature drifts
Left: Global mean temperature changes from first year
Below: Global mean temperature change between years 1 and 30
Control
With z~
Reduction in drift by 5-10% with z~

25. Ratio of keff to explicit diffusivity kexp in control and z~ runs
Mixing diagnostics
Diagnosed effective diapycnal diffusivity with z~ is generally 5-10% lower than that in control.
Ratio of effective diffusivity keff in integration with z~ to keff in control
Ratio of keff to explicit diffusivity kexp in control and z~ runs

26. Why might z~ not reduce numerical mixing more?
Distortion of coordinate surfaces puts more stress on accuracy of horizontal advection scheme: is horizontal advection projecting significantly onto diapycnal direction?
The isoneutral mixing scheme has a limiter on the angle of neutral surfaces from horizontal: is this being exceeded?
z~ will not affect the poorly resolved, but persistent (i.e. timescale longer than ~10 days), features at higher latitudes, which are known to produce numerical mixing (Megann and Storkey, 2019)

27. Possible approaches to improving z~
Use higher-order horizontal tracer advection scheme to reduce truncation errors
Jerome Chanut recommends switching to the Griffies triads for isoneutral diffusion: better separation of isoneutral and dianeutral directions.
Increase the viscosity to suppress the noise-like ”eddies”
Try a longer cutoff timescale for z~ (default is 5 days)

28. Further experiments
An ensemble of experiments is currently underway to attempt to optimise z~ configuration. Parameter changes include:
Changing horizontal advection from 2nd to 4th order;
Increasing value of background biharmonic viscosity;
Changing from constant biharmonic viscosity to Smagorinsky scheme (Cmsmag=4);
Change from Redi isoneutral mixing to Griffies;
Changing vertical advection from 2nd to 4th order;
Increasing cutoff time for z~ scheme from 5 days to 10 days;

29. Preliminary results from sensitivity experiments: temperature drifts
Temperature changes between years 1 and 30
Improvement from increasing accuracy of horizontal advection is larger than that from z~ itself!
2nd order
4th order
Better!

30. Preliminary results from sensitivity experiments: diagnosed diffusivity
Ratio of keff to keff in control: z~ with 4th order advection
Ratio of keff to keff in control
The combination of z~ with 4th order horizontal advection is the least diffusive so far, with reductions in keff of >20%.

31. Summary
A tidally-forced GO8p0.1 NEMO ¼° global configuration has been integrated for 30 years, and is currently being evaluated.
A 30-year integration with z~ has been completed. Using z~ “straight out of the box”, it seems to be doing what it should...
… but in retrospect it would have been surprising if it had led immediately to big reductions in numerical mixing.
z~ significantly stresses the numerical schemes, so a set of modifications to the numerics is being assessed to get the full benefits. Preliminary results with 4th order horizontal advection are promising…

32. Milestones achieved
M1: Tides working correctly in model and 30-year test integration completed.
Completed March 2019
M2: z-tilde successfully implemented in global NEMO and 30-year test integration completed
Completed February 2019

33. Issues
Initial tidally-forced configuration was unstable, with velocities > 10 ms-1 and SSH > 15m. Solved by deepening all the shelves to 23 m.
Tidal diagnostics provided with NEMO v4-beta are not complete, and a development branch of this code is being assessed.
First z~ integration crashed: eventually a bug in iceberg scheme was found and corrected.
z~ by itself only has a relatively small effect on numerical mixing. A suite of sensitivity experiments are being carried out to find an optimal combination of numerical settings.

34. Projected benefits to CMEMS
Demonstration that tidal forcing can be applied, in a relatively straightforward way, to a 20-25km global NEMO.
Analysis of effects of tidal motions on watermass properties and shelf transports in this configuration.
Demonstration of working z~ scheme in global eddy-permitting ocean configuration.
Confirmation that z~ does reduce numerical mixing, and that its performance can be further improved by tuning other parameters.

35. Communications
PI Megann presented a talk on “Adventures in 1/4° NEMO land” at the Joint Marine Modelling Programme (JMMP) Team Meeting, NOCS, 2/3. October 2018, including a summary of results to date from RENUMERATE project
PI Megann presented a talk on “First steps with z-tilde in a 1/4° global NEMO” at the DRAKKAR workshop, Grenoble, January 2019
A summary of the results was presented as a highlight at the NOC Marine Systems Modelling group Annual Science Meeting in Winchester, 6/7 March 2019