1. Idealised circular basins:
The effect of bathymetry on barotropic geostrophic adjustment in an idealised Arctic Ocean basin. Maria Luneva, Andrew Willmott and Miguel Angel Morales Maqueda
Arctic Basin
Idealised circular basins:
R=15o, Depth=3000m
Ridge , step and slope
topography anomalies

2. Background Short history of question:
F-plane : Rossby (1937,1938), Johnson (1985), Gill et al. (1986).
F-plane: Existence of steady state solution for escarpment in infinite basin (Willmott and Grimshaw, 1991,1992).
Circular basin, flat bottom: Analytical steady state solution and laboratory experiments for 2-layer fluid with initial step in thickness of the top layer (Wake et al., 2004, 2005).
Circular basin with polar plane: Free waves in circular basin with flat topography (LeBlond, 1964); Harlander, 2005).

3. Model: NEMO – filtered free surface algorithm, 10km resolution.
Goal: To understand basic aspects of the barotropic adjustment problem in the Arctic using very simple idealised configuration.
Model: NEMO – filtered free surface algorithm, 10km resolution.
Initial condition: sea surface height step
A) – Y<0: SSH=0.4m,
Y>0: SSH=-0.4m
B) bump 0.4m
Coriolis force: f-plane and polar Coriolis.

4. F –plane: Ridge and step topographies + step in SSH, numerical solution:
Initial adjustment : fast barotropic waves
After 3 days of integration, a quasi-steady 4-gyre circulation is formed.
The strength of circulation is inversely proportional to the depth.
In the step topography case, the solution in each semicircle does not depend on the magnitude of the step.
Amplitude is very slowly decaying with time, just 5% per year.
Steady state solution for semi-circular basin?
(a)-(b) SSH on day 3 (colour) and day 710 (contours) for step and ridge topography
(c) Slice of SSH
black : X=80 step
red : X=217 step
green: X=80 ridge
blue : X=217 ridge
l blue : SSH*2/3 X=217 step
(d) Decay of the amplitude of the circulation with time (depends on time and spatial steps in filtered free surface algorithm). a b

5. F –plane: Smooth step (200 m) + step in SSH, numerical solution:
Initially 2 gyre system is formed (as in flat bottom case, see Wake et al., 2004)
This 2 gyre is unstable.
Final solution is a 4 gyre system, as before, plus a system of barotropic Rossby waves propagating over the smoothed step.
Analytical solution for 2 layer baroclinic case and flat bottom from Wake et al. (2004).

6. F-plane: Flat, step and ridge topographies
F-plane: Flat, step and ridge topographies. Initial condition: Radially symmetric bump in SSH
Steady states :
1. Flat bottom: symmetric gyre.
2. Ridge topography; 4 gyre x-y axisymmetric system, second mode in radial coordinate
3. Step topography: flow is the same as in ridge case but with amplitude dependent on the basin depths.
4. Initially formed 1 gyre system splits into steady 2-gyre system on the left part of basin and topographical waves on the right part.

7. F –plane: step in SSH, semicircular basin analytical solution:
Wake et al, J. Fluid. Mech., 2004, 2005:
2-layer baroclinic problem, circular basin, flat bottom
Initial condition : a step in layers heights.
The analytical solution for the steady state is 2-gyre system .
Unsteady equation for SSH:
In the semi-circle case:
In steady case for dimensionless
variables:
problem reduces to equation:
with coefficients:
with boundary conditions:
Final solution is similar to:
Let us look the solution in the form:

8. Initial bump in SSH – 2 radial gyres steady state solution.
F –plane: step in SSH, semicircular basin analytical solution for initial step in SSH:
Asymptotes at
For large S solution is proportional to S-2, :
Thus amplitude of the steady state solution is inversely proportional to the ocean depth,
η~ S-2=R2/(gH), η~H-1
Comparison with numerical solution
numerical solution: prediction
(η/ η0)max= (η/ η0)max=0.0265/S2=0.0517
Initial bump in SSH – 2 radial gyres steady state solution.
3- mode composite and
2,4,6 modes

9. F –plane: step in SSH, semicircular basin analytical solution:
The analytical solutions in a semicircular basin with flat bottom describe well the solutions in the circular basin with a step topography or middle ridge.
The solutions in each half-basin with either step or ridge topographies are independent.
The analytical solution is simplified for the case of large S ( radius of basin is comparable with Rossby radius) and this asymptotic is proportional to S-2 .
The amplitude of the solution is nearly inversely proportional to the depth of the ocean.
If the topography anomaly is small, initial stage – 2gyre system is as in the flat bottom case. The asymptotic solution for longer times are a 4-gyre system.
Questions:
What is the actual solution for the case of a semicircular basin with a flat bottom?
How the solutions look like in the polar plane?

10. F-plane :sloping topography rising/deepening with negative X.
Initial 2-cell circulation is unstable.
Initial disturbance tends to shift cyclonically in shallower slope and clockwise in opposite case.
Left part of basin: topographic basin scale topographic Rossby waves propagating along the slope with shallower water to the right.
Right part of basin:
superposition of steady 2-gyre system and weaker oscillations.

11. Some curious topography..
Step topography with semiconical canyon.
4 gyres system with wave channel
at the middle.
2-gyre steady state solution tends to establish over flat parts of basin.
SSH evolution during one year of modelling time

12. Effects of polar Coriolis force flat topography.
Polar Coriols force: F=2Ω cos(θ), θ<<1
Polar plane: F≈2Ω (1-θ2/2), θ=90-φ
Free waves in polar plane: analytical solutions: LeBlond (1964), Longet-Higgins (1964), Harlander (2005).
LeBlond solution: first mode T = 120 days.
Initially the solution is close to Wake et al (2004) 2-cell gyre.
Numerical solution : clockwise propagating Rossby waves with period of 120 days.
Asymmetry- presence of higher modes.

13. Extended analysis of LeBlond’s solution:
LeBlond, 1964 : approximate solution for the free waves in polar circular basin:
Roots of Jk(k)
roots of Bessel function linearly depend on s for s=1:5 . Approximate dispersion relation:
Phase velocity is always westward :
Group velocity
Periods for free zonal waves

14. Extended analysis of LeBlond solution:
Group and phase velocity zonal modes: M=29, ε=0.67:
Absolute value of dimensionless
phase velocity
Dimensionless group velocity
always directed eastward!

15. Polar Coriolis force – “delta-plane”: step and ridge topographies F=2Ω cos(θ), θ<<1 : F≈2Ω (1-θ2/2), θ=90-φ
As in F-plane circulations in step and ridge topography are very similar.
In the case of a step topography, circulation is not symmetric and stronger at the shallow part.
Initial 4-cell circulation corresponds to the steady state solution in semi-circular basin.
This 4 gyre solution is unstable: Gyres tend to propagate anti-cyclonically as Rossby waves.
In the early stages, length scales of Rossby waves are about the basin scales (n=2), even gyres is weakening.
Circulation losses axial symmetry and becomes radially symmetric.

16. Polar Coriolis force: step and ridge topography.
After 3 months, the circulation is completely radially symmetric for ridge topography and with amplification in amplitude in step topography.
length-scales become much shorter and non-uniform;
Phase velocity is clockwise;
Group velocity is counter-clockwise;
Maximum is slowly moving counter-clockwise;
Solutions in either half of the basin correspond to solution on a semicircle.
Colour: SSH at 08 April and 25 October;
Contours: AAH 15 days later.

17. Polar Coriolis force: step and ridge topography
After 2 years of integration, the amplitude of waves remains essentially unchanged.
Maximum anomaly propagates counter-clockwise.
Dominating frequencies for SSH: 106, 120 and 136 days.
Dominating frequency for KE: 191, and 60 days.

18. Polar Coriolis force: step and ridge topography.
Cross-sections along X=230 on different dates:
Black- day 200, red –day 205
Green- day 600, blue-day 605
Black arrow indicates the direction of phase velocity and red arrow – propagation of wave maximum.
Estimate of group velocity cg=8o/400days=2.2km/day~ 2.5cm/s
Very rough estimate of phase velocity at peak:
Cp=4o/1month~16cm/s
Further analysis is necessary:
Is this a combination of Rossby waves and low-frequency Kelvin waves? or
Wave packets of incident and reflected Rossby waves with both cyclonic and anticyclonic group velocity?

19. Polar Coriolis force. Flat, step and ridge topography
Polar Coriolis force. Flat, step and ridge topography. Initial condition – the radially -symmetric bump in SSH
A steady state solution exists only for the flat bottom case.
The initial phases for step and ridge topographies are similar to the steady state solutions for the f-plane case– 4 gyre system.
Further, this gyres tend to rotate clockwise which leads to shifting of the gyres.
Late stage is the arrested 2 dipole system.
Axisymmetric solutions.

20. Conclusions
In the simple barotropic case, analytical solutions for the semicircular basin are derived for the case of uniform rotation.
These solutions are used for validation of the numerical simulations.
For the symmetric ridge and step topography, the circular basin solutions on each half of the basin are the same as the solutions of the semicircular problem.
The initial adjustments on the polar-plane problems are similar the solutions on the f- plane.
On longer time scales, however, Rossby waves are responsible for further evolution of the flow, the nature of which needs to be further examined.
Existing theory for zonal Rossby waves on the polar plane predicts only eastward propagation of energy and can not explain westward wave reflection.

21. The effect of the bathymetry on the propagation of Atlantic water anomaly
Topography: circle basin with/without shelf, slopes:
Radius of basin: R=15o,
depth of shelf =200m,
width of shelf=200km,
Width of asymmetric semicircle shelf 1/2R.
Stratification: 3 layer
densities : averages over 0-200m, m and m of Levitus data .
Temperature is recalculated form density for the salinity 35 PSU.
Model: NEMO with vertical full steps, nonlinear filtered free surface
Resolution: 0.1o, 21 vertical levels.
NO OTHER FORCING!
Wide asymmetric shelf

22. The effect of the bathymetry on the propagation of Atlantic water anomaly.
Due to asymmetry of shelf initially symmetric anomaly splits onto two branches:
on the deep side of basin cold anomaly propagates counter-clockwise and
shallow branch in opposite direction with velocities exceeding 10/15 cm/s.
Wide asymmetric shelf
Figure 1. Upper corner: Initial temperature across the basin and bottom topography: flat bottom, symmetric narrow
shelf, asymmetric shelf. Surface temperature after 1 year for these cases :contours indicate the location
of shelf (200m deep) and initial location of Atlantic water anomaly.

23. Experiment with flat bottom topography
Two processes are evident from this figure:
eddy restratification of the core
and entrainment of upper/lower water to the Atlantic waters.
Temperature at the crossection throw the basin at different time
(colour- initial condition,
black contours – after 0.5 year and red –after 1 year) and temperature at different depths.

24. Propagation of Atlantic water anomaly in the circle basin with wide shelf
After 2 year of modelling time:
anomaly at the deep part of basin propagates eastward onto 1000km
cyclonic circulation along shelf with velocities about 10cm/s
SST and velocity at different time (every 6th arrow is shown).

25. Propagation of Atlantic water anomaly in the circle basin with wide shelf
Temperature at 100 and 300m , experiment with wide shelf
Water propagating cyclonically along the shelf exhibit intensive vertical mixing due to eddy activity.
As the result, signature of warmer water appears at the depth of m.

26. Small anomaly: narrow shelf, wide shelf, wide shelf+slope
Circulation consists of :
cyclone at the location of initial anomaly
reverse current in lower layer
and cyclonic circulation along the shelf.
Asymmetry of shelf causes propagation of colder water in upper layer cyclonically.
Along-shelf currents are twice stronger in the presence of slope.
Temperature and velocity at surface and 300m after 4 years of modelling time.
Upper panel: velocity scale=2cm/s, lower panel scale=0.5cm/s

27. Conclusions
Eddy restratification and entrainment are important processes in adjustment in the Arctic basin
Without any additional forcing the presence of shelf and slope can and Atlantic layer anomaly can induce the cyclonic circulation.
The asymmetry of the shelf amplify this cyclonic circulation
Further work:
To estimate eddy fluxes and how those effects the restratification.
dN2/dt=…
To examine the physical mechanisms of intensive currents along the asymmetric slopes.