The Barotropic Vorticity Budget of Subtropical and Subpolar Gyres

Su et al. (2018)

The barotropic vorticity budget of the Weddell Gyre Andrew Styles Department of Physics | University of Oxford Gyres

Gyres are basin scale circulations in the world ocean

Features • Significant meridional transport (~10 Sv)

• Closed by intense western boundary currents COMET - http://meted.ucar.edu/

• Often driven by a wind stress curl Error in HadGEM3-GC3.1

Scaife et al. 2011 In the North Atlantic (Subtropical) • Systematic error off Western coast

• ~7℃ error in 1 degree HADGEM3 model

• Error is reduced in higher resolution models

• Becomes highly sensitive to bathymetry - Mean winter SST error (℃) ‘bathymetric steering’ HadGEM3 Error in HadGEM3-GC3.1

In the Southern Ocean (Subpolar) • Poor representation of ACC transport at Jonathan et al., 2020 EGU General Assembly 2020 higher resolutions

• ACC transport is weakest at 1/4°

• Attributed to slumping of isopycnals near southern boundary

• Consequently related to strengthening of the Weddell Gyre (WG) Evolution of ACC transport for R1, R4 and R12 HadGEM3-GC3.1 models Error in HadGEM3-GC3.1

Figure from D. Storkey In the Southern Ocean (Subpolar) • WG strengthening occurs in the uncoupled global model also.

• WG transport increases with resolution Resolution: 1° Transport ∼17 Sv • WG extent increases with resolution

• WG expands into Drake passage at 1/4° and 1/12° resolution Stream function of Weddell Gyre in uncoupled model (10 yr average) White contours show bathymetry (every 700 m) Error in HadGEM3-GC3.1

Figure from D. Storkey In the Southern Ocean (Subpolar) • WG strengthening occurs in the uncoupled global model also.

• WG transport increases with resolution Resolution: 1/4° Transport ∼ 31 Sv • WG extent increases with resolution

• WG expands into Drake passage at 1/4° and 1/12° resolution Stream function of Weddell Gyre in uncoupled model (10 yr average) White contours show bathymetry (every 700 m) Error in HadGEM3-GC3.1

Figure from D. Storkey In the Southern Ocean (Subpolar) • WG strengthening occurs in the uncoupled global model also.

• WG transport increases with resolution Resolution: 1/12° Transport ∼ 28 Sv • WG extent increases with resolution

• WG expands into Drake passage at 1/4° and 1/12° resolution Stream function of Weddell Gyre in uncoupled model (10 yr average) White contours show bathymetry (every 700 m) Error in HadGEM3-GC3.1

Figure from D. Storkey Additional model behaviours

• Adding partial slip boundary condition helps to reduce WG transport

Resolution: 1/4° • Using Gent-McWilliams reduces the Transport ∼ 31 Sv transport but reduces eddy energies

• Adjusting the scheme for the Coriolis force affects the WG transport Project Aims

Project aims

• Study the vorticity budget of the Weddell Gyre Main focus of this presentation • Compare results with a simple box model 휻 (GYRE PISCES)

• Identify any increased forcing or reduced dissipation at higher resolutions Vorticity and Gyres

Introducing the barotropic vorticity equation

휕휁 휏 1 푠 ෠ 2 = −∇ℎ ⋅ 푓푼 − ∇ℎ ⋅ 휁푼 + ∇ × ⋅ 푘 − 푟휁 + 휅∇ 휁 + ∇푃 × ∇퐻 휕푡 휌퐻 휌0

Obtained by taking the curl of the barotropic momentum equation

푼 is the depth-integrated velocity 휁 is the depth-integrated relative vorticity Vorticity and Gyres

Introducing the barotropic vorticity equation

휕휁 휏 1 푠 ෠ 2 = −∇ℎ ⋅ 푓푼 − ∇ℎ ⋅ 휁푼 + ∇ × ⋅ 푘 − 푟휁 + 휅∇ 휁 + ∇푃 × ∇퐻 휕푡 휌퐻 휌0

Time Coriolis forcing Advection Wind stress curl Bottom Lateral Bottom evolution = 훽푉 of relative drag diffusion pressure of for vorticity (linear) torque relative incompressible Corresponding vorticity flow codes for model vorticity trends TOT PVO ADV WND FRC LDF HPG Vorticity and Gyres

Different balances characterise different types of Gyres

휕휁 휏 1 푠 ෠ 2 = −∇ℎ ⋅ 푓푼 − ∇ℎ ⋅ 휁푼 + ∇ × ⋅ 푘 − 푟휁 + 휅∇ 휁 + ∇푃 × ∇퐻 휕푡 휌퐻 휌0

For example a steady, linear flow in the absence of bathymetry and diffusion

This is the vorticity balance of the Stommel (1948) gyre. Local balances

We can identify the leading order balances in local areas (we will come back to this plot later) Two largest contributions

HPG – PVO

0.87

Fraction of the vorticity budget occupied by the two largest terms Contour integrations

Local balances are useful but it can be difficult to determine how the gyre transport is affected.

Instead then, we choose to integrate over the area enclosed by the streamlines.

퐼 휓 = ׭ ∇ × 푴 ⋅ 풌෡ 푑퐴 = ׯ 푴 ⋅ 풅풍 퐴휓 휓

Vorticity diagnostic associated Work done per unit mass on a with momentum diagnostic fluid column circulating the (force per unit mass) 푴 gyre once Contour integrations

Before we get to results, let’s summarise the full procedure

1. Depth integrate momentum trend 풎

Depth integrate

GYRE box model is used for illustrative purposes Contour integrations

Before we get to results, let’s summarise the full procedure

2. Take the curl of 푴 using 푀푢 and 푀푣

Curl

GYRE box model is used for illustrative purposes Contour integrations

Before we get to results, let’s summarise the full procedure

3. Interpolate ∇ × 푴 ⋅ 풌෡ to a fine grid to minimise edge effects when integrating

Interpolate to 1/12°

GYRE box model is used for illustrative purposes Contour integrations

Before we get to results, let’s summarise the full procedure

4. Mask values outside of a given streamline 휓 (using Scikit-image)

Mask outside of streamline 휓

GYRE box model is used for illustrative purposes Contour integrations

Before we get to results, let’s summarise the full procedure

5. Integrate over enclosed area Integrate 퐼(휓)

GYRE box model is used for illustrative purposes Contour integrations

Before we get to results, let’s summarise the full procedure

1. Depth integrate momentum trend 풎

2. Take the curl of 푴 using 푀푢 and 푀푣 3. Interpolate ∇ × 푴 ⋅ 풌෡ to a fine grid to minimise edge effects when integrating 4. Mask values outside of a given streamline (using Scikit-image) 5. Integrate over enclosed area Weddell Gyre results

1/4°

Discontinuity is due to enforced contour As the Weddell criteria Gyre circulates in a clockwise direction ׯ 풅풍 is in the opposite direction to the Value of flow streamline integrated inside of Weddell Gyre results

This is a 1/4° stacked area plot so this arrow indicates the value for HPG

High 휓 contours are smaller and lie in the Gyre interior Low 휓 contours are large and lie close to the ACC and Antarctic Weddell Gyre results

Bottom pressure 1/4° torque is a significant drag

휷푽 is the dominant opposing force?

Wind stress curl spins the Gyre up Why is PVO so big?

Analytically we would expect the Coriolis contribution to be zero

෡ ෡ 퐼푃푉푂 휓 = ඵ 훁 × −푓풌 × 푼 ⋅ 풌 = − ඵ 훁풉 ⋅ 푓푼 푑퐴 퐀흍 푨흍

Using the divergence theorem we would expect As the flow is parallel to the streamlines

퐼푃푉푂 휓 = − ර 푓푼 ⋅ 풏ෝ 푑푠 = 0 흍 Why is PVO so big?

On the model (C) grid would we expect the same?

Data points lie on a staggered grid

• Scalar values (e.g. 푆푆퐻) on T points

• Velocities on U and V points

• Vorticities and stream functions (e.g. 푓 and 휓 ) on F points Why is PVO so big?

Calculating the Coriolis force is not trivial

The exact method depends on which Coriolis scheme is used.

In this case the EEN (energy and enstrophy An example of a conserving scheme) is used. North-East F triad

EEN relies on thickness weighted triads of F points. Why is PVO so big?

The curl of the depth-integrated Coriolis force is not simple …

Look at two simple cases to interpret this Simple case 1: Flat topography

With flat topography (no partial cells)

• Integrated over a “rectangular” stream function

• Velocity equals 푈 everywhere on all sides except western boundary

• Simple intensification on western boundary, 푈 + Δ Simple case 1: Flat topography

In this case

1 퐼 = 푓푁퐸 − 푓푁퐸 + 푓푆퐸 − 푓푆퐸 Δ 푃푉푂 12 푖0+1,푗0+푀−1 푖0+1,푗0+2 푖0+1,푗0+푀−1 푖0+1,푗0+2

Additional 훽 effect on intensified current

An example of a significant non-zero integral (∼ 10 푚3푠−2) Simple case 1: Flat topography

Verified in the GYRE box model Dashed line is my estimate of

PVO integral

]

2 −

푠 Orange curve is 3

푚 model output [

휓 Decomposition of PVO contour integral in flat GYRE model. Derived form of PVO matches model output exactly Simple case 1: Flat topography

This breaks down when topography is introduced Dashed line is my estimate of

PVO integral

]

2 −

푠 Orange curve is 3

푚 model output [

Estimate is missing a feature for the subtropical gyre 휓 Decomposition of PVO contour integral in sloped GYRE model. Derived form of PVO deviates severely Simple case 2: F plane with topography

F plane (훽 = 0) with varying topography (partial cells) Cell thicknesses at lowest level 푒3 1 − 훼 푒3 푒3 1 + 훼 Assuming a gentle slope (훼 ≪ 1)

In this case the PVO vorticity trend is

푓 훼 = 0 푤 + 푤 − 푤 + 푤 ∼ 10−10 푚푠−2 3 푖+1,푗,0 푖+1,푗+1,0 푖푗,0 푖,푗+1,0

A numerical vortex stretching on down-slope gradients in 푤 Why is PVO so big?

The PVO integral has two contributions:

• Numerical beta effect on intensified currents Can calculate explicitly

• Numerical vortex stretching of down-slope gradients in 푤 Currently unable to calculate accurately

Which of these dominates in the Weddell Gyre? Num. Vortex stretching in the Weddell Sea

1/4°

Only ∼ 20% of PVO is accounted for by intensification Bottom pressure torques are strongly This weakly correlated with implies the PVO (푟2 = 0.9) remainder of PVO is due to numerical vortex stretching. Num. Vortex stretching in the Weddell Sea

Two largest contributions

HPG – PVO

0.87

Fraction of the vorticity budget occupied by the two largest terms Summary

The Weddell Gyre has a strong dependence on model resolution

Studying the barotropic vorticity balance at 1/4° identifies Wind stress curl and PVO as the leading order balance.

The PVO contribution has at least two numerical contributions due to Intensification of the current Numerical vortex stretching

In the 1/4° model it appears numerical vortex stretching dominates Future work

• Analyse vorticity balances of Weddell Gyre at 1° and 1/12°

• Attempt to recreate the numerical vortex stretching contribution

• Look at similar data sets with roughened/smoothed bathymetry

• Determine if these numerical factors are “real” and affecting the circulation. Extra Slides Weddell Results (R4) HPG

PVO LDF

ADV ZDF

WND (upper 100 m of ZDF) GYRE results Gyre results

Flat GYRE configuration R4