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Potential Vorticity diagnostics based on balances between1
volume integral and boundary conditions2
Yves Morel1,⇤, Jonathan Gula2,⇤, Aur´elien Ponte2,⇤
3
Abstract4
Taking advantage of alternative expressions for potential vorticity (PV) in5
divergence forms, we derive balances between volume integral of PV and6
boundary conditions, that are then applied to practical computations of PV:7
8
•we propose a new method for diagnosing the Ertel potential vorticity9
from model output, that preserves the balances;10
•we show how the expression of PV can be derived in general coordi-11
nate systems. This is here emphasised with isopycnic coordinates by12
generalising the PV expression to the general Navier-Stokes equations;13
•we propose a generalised derivation for the Haynes-McIntyre imper-14
meability theorem, which highlights the role of the bottom boundary15
condition choice (e.g. no-slip vs free-slip) and mixing near the bottom16
boundary for the volume integral of PV.17
The implications of balances between volume integral of PV and boundary18
conditions are then analysed for specific processes at various scales:19
•at large scale, we show how an integral involving surface observations20
(derived from satellite observations) is linked to the integral of PV21
⇤Corresponding author
Email addresses: yves.morel@shom.fr (Yves Morel),
jonathan.gula@univ-brest.fr (Jonathan Gula), aurelien.ponte@ifremer.fr
(Aur´elien Ponte)
1LEGOS, Universit´e de Toulouse, CNES, CNRS, IRD, UPS, Toulouse 31400, France
2LOPS, Universit´e de Brest, CNRS, Ifremer, IRD, IUEM, Brest 29280, France
Preprint submitted to Ocean Modelling April 23, 2019
within a layer (between two isopycnals). This surface integral can be22
calculated for models and observations and can be used for validation;23
•at mesoscale or sub-mesoscale, we analyse the relationship between net24
PV anomalies and net surface density anomalies for idealised vortices25
and 2D fronts. This can help determining vortex or jet structures for26
idealised studies or empirical methodologies;27
•we also confirm and integrate previous results on the modification of28
PV within a bottom boundary layer into a single diagnostic taking29
into account the e↵ect of density and velocity modifications by dia-30
batic processes along the topography and diapycnal mixing within the31
boundary layer.32
Keywords: Potential vorticity, boundary conditions, general circulation,33
vortex, fronts, boundary layers.34
1. Introduction35
It is well known that Ertel’s Potential Vorticity (PV, see Ertel, 1942)36
is an important quantity when studying the circulation at all scales in geo-37
physical fluids: the conservation property of PV -in adiabatic evolution- and38
the inversion principle (the geostrophic velocity field can be inferred from39
the PV field and boundary conditions) are key principles to interpret the40
ocean dynamics (see Hoskins et al., 1985; McWilliams, 2006, and section 241
for more details). Conservation and inversion of PV are the basis of the42
quasigeostrophic (QG) model (Pedlosky, 1987) that has been successfully43
used in pioneering studies aiming at understanding and modelling the ocean44
circulation from basin gyres (Rhines and Young, 1982a,b; Luyten et al.,45
2
1983; Holland et al., 1984; Rhines, 1986; Talley, 1988; Marshall and Nurser,46
1992) to current instabilities (Charney and Stern, 1962), geophysical turbu-47
lence (McWilliams, 1984) and mesoscale dynamics (McWilliams and Flierl,48
1979; Sutyrin and Flierl, 1994).49
In the QG framework, PV is related to the streamfunction by a linear50
elliptic di↵erential operator (Pedlosky, 1987; Cushman-Roisin and Beckers,51
2011), which has several important consequences. First, boundary condi-52
tions impose important dynamical constraints too. In a QG framework53
Bretherton (1966) has shown that surface or bottom outcropping of isopyc-54
nic surfaces is dynamically similar to a shallow layer of high PV anomaly (in55
practice a Dirac delta sheet), whose strength can be related to the density56
anomaly. This has led to the generalised surface quasigeostrophic (SQG)57
model (Held et al., 1995; Lapeyre, 2017). Lateral boundaries can be impor-58
tant too for the inversion of PV. In the QG or SQG framework, it has been59
shown that the velocity field away from a region of PV anomalies decreases60
slowly -as the inverse of the distance from the region- unless PV and surface61
density satisfy an integral constraint (Morel and McWilliams, 1997; Assassi62
et al., 2016). In models, practical inversion of PV, with given surface and63
bottom density fields, is often done considering biperiodic domains (Lapeyre64
et al., 2006; Wang et al., 2013), which can lead to discrepancies if the latter65
constraint is not satisfied.66
Second, since the relationship between PV and the circulation is linear67
at first order (QG and SQG), the balance between smoothed/averaged fields68
is preserved, provided averaging is done using a linear convolution.69
Moreover, PV concept is also useful for forced dissipative dynamics.70
For instance, diapycnal mixing does not change the volume integral of PV71
within a layer bounded by isopycnic surfaces, which shows that PV can72
3
only be diluted or concentrated when the layer respectively gains or looses73
mass (Haynes and McIntyre, 1987, 1990). The influence of viscous surface74
or bottom stress on the PV evolution has also been analysed theoretically75
(Thomas, 2005; Taylor and Ferrari, 2010; Benthuysen and Thomas, 2012,76
2013). Thus, the consequences of diabatic e↵ects on the ocean dynamics77
can again be analysed and interpreted in terms of PV modification from78
basin scales (see for instance Hallberg and Rhines, 1996, 2000; Czaja and79
Hausmann, 2009) to meso and submesoscales (see for instance Morel and80
McWilliams, 2001; Morel et al., 2006; Morel and Thomas, 2009; Rossi et al.,81
2010; Meunier et al., 2010; Thomas et al., 2013; Molemaker et al., 2015;82
Gula et al., 2015, 2016, 2019; Vic et al., 2015; Giordani et al., 2017).83
To conclude, the ocean circulation and PV are linked and calculating PV84
at all scales under adiabatic or diabatic conditions is thus of considerable in-85
terest for geophysical fluid dynamics. In QG or SQG models, it is possible to86
ensure consistent balances between circulation, PV and surface, bottom and87
lateral boundary conditions, from local to averaged fields. In more complex88
models, PV calculation involves many velocity and density derivatives, in89
particular in non-isopycnic models, and keeping the link between averaged90
PV and averaged circulation implies to find a consistent calculation of PV.91
If several studies have used diagnostics involving PV, they remain rare and92
none have discussed the PV calculations in details, in particular to evaluate93
if the relationships between PV and boundary conditions are maintained94
and if averaging can be done consistently.95
The Bretherton principle (Bretherton, 1966) has been recently revisited96
and extended by Schneider et al. (2003) who generalised the concept of PV97
to take into account the dynamical e↵ect of outcropping for the general98
Navier Stokes equations. To do so, they used the alternative divergence99
4
form for the expression of PV (Vallis, 2006). In this paper, we show how100
this divergence form of PV naturally leads to general constraints on volume101
balances of PV and boundary conditions (section 3). These follow from the102
definition of PV and are independent of the dynamics (adiabatic or diabatic)103
of the flow. In section 4 we show that the divergence form also makes PV104
computations easier and consistent, in the sense that balances are automati-105
cally preserved when integrating PV (a consequence of the divergence form).106
We then propose several frameworks, involving dynamics at di↵erent scales,107
to discuss the generalised constraints between PV and surface, bottom or108
lateral boundary conditions (section 5). We summarise and discuss our re-109
sults in the concluding section. Section 2 summarises basic definitions and110
properties of PV which are not new and can be skipped by readers familiar111
with PV.112
2. Reminders on potential vorticity113
2.1. Definition of Ertel potential vorticity114
Ertel (1942) defined Potential Vorticity as:115
PV
Ertel ⌘(~
r⇥~
U+~
f).~
O⇢
⇢
=(~
r⇥~
Ua).~
O⇢
⇢(1)
where ~
Uis the fluid velocity field in the reference frame of the rotating Earth,116
⇢is the potential density (in the ocean and entropy in the atmosphere),117
~
Ua=~
U+~
⌦⇥~ris the absolute velocity, where ~
⌦=(0,⌦y,⌦z)isthe118
rotation vector of the Earth, ~ris the position relative to the Earth center119
and ~
f=(0,f
y,f
z)=~
r⇥(~
⌦⇥~r)=2~
⌦(see Fig. 1). Note that ~
fis120
5
fixed but its components in some coordinate system (spherical coordinates121
for instance) can vary with position. The minus sign on the left-hand side of122
(1) is so that PV is generally positive for gravitationally stable - low Rossby123
number flows in the northern hemisphere.124
Figure 1: General Earth referential.
In the ocean, the Boussinesq approximation is typically valid and ~
O⇢/⇢125
can be replaced by ~
O⇢/⇢0,where⇢0is a mean oceanic density. ⇢0can then126
be omitted from the definition of PV and we can use:127
PV
Ertel =(~
r⇥~
U+~
f).~
O⇢
=(~
r⇥~
Ua).~
O⇢(2)
We retain this definition for PV as it leads to clearer expressions for the128
calculations we present and the formulas we obtain. This approximation is129
6
however not necessary and all the following results are valid provided ⇢is130
replaced by G(⇢)=log(⇢) (see Appendix B).131
2.2. Properties132
2.2.1. Conservation133
The non-hydrostatic Navier-Stokes equations (with Boussinesq approxi-134
mation) are:135
d
dt ~
U+~
f⇥~
U=~
rP
⇢0~g⇢
⇢0
+~
F
div(~
U)=0
d
dt⇢=˙⇢(3)
where ~
U=(u, v, w)isthevelocityfield, d
dt =@t+⇣~
U.~
r⌘,~
f=(0,f
y,f
z)136
is the Coriolis vector, Pis the pressure, ⇢is the potential density and137
~
F=(Fx,F
y,F
z) and ˙⇢are terms associated with diabatic processes for138
momentum and density fields.139
The Lagrangian evolution of Ertel PV can be derived from Eq. 3:140
d
dtPV
Ertel =(~
r⇥~
F).~
O⇢(~
r⇥~
U+~
f).~
O˙⇢(4)
As shown by Ertel (1942), PV
Ertel is thus conserved in regions where diabatic141
processes are negligible.142
The evolution/conservation of PV following fluid particles is a major143
constraint for geophysical fluid dynamics (Hoskins et al., 1985). To study144
geophysical fluids, simplified forms of Eq. 3 are sought which conserve a sim-145
plified expression for PV (White et al., 2005). This is the case for instance for146
7
quasigeostrophic or primitive equations (Pedlosky, 1987; Cushman-Roisin147
and Beckers, 2011; McWilliams, 2006). For the primitive equations, the hy-148
drostatic approximation is assumed and fyis neglected, PV can be written149
(White et al., 2005):150
PV
PE =(@xv@yu+fz)@z⇢+@zv@
x⇢@zu@
y⇢(5)
where fzis the (local) vertical component of the Coriolis vector and is called151
Coriolis parameter.152
The Lagrangian conservation of PV
PE is more conveniently derived, and153
achieved in numerical models, using density ⇢instead of the geopotential154
vertical coordinate z. This has been one of the motivation for the develop-155
ment of isopycnic coordinate ocean models (see for instance Bleck et al.,156
1992; Hallberg, 1997). Using isopycnic coordinate, PV
PE can be written157
(Cushman-Roisin and Beckers, 2011):158
PV
PE =⇣+fz
h(6)
where ⇣=(@xv@yu)|⇢is the relative vorticity, now calculated using159
horizontal velocity components along isopycnic surfaces and h=@⇢zis a160
measure of the local stratification. We will see below how the expression of161
PV can be easily derived in isopycnic coordinates for the full Navier-Stokes162
equations (including terms coming from all components of the Coriolis vector163
and non-hydrostatic e↵ects).164
2.2.2. Inversion165
If (cyclo)geostrophy is assumed, the velocity field and stratification can166
be calculated from the PV and are associated with the balanced dynamics167
8
(Hoskins et al., 1985; Davis and Emanuel, 1991; McIntyre and Norton, 2000;168
Morel and McWilliams, 2001; Herbette et al., 2003, 2005). The PV of a fluid169
at rest and with a horizontally homogeneous stratification is not null. The170
potential vorticity anomaly (PVA) is defined as the di↵erence between total171
PV and a reference PV associated with a state of rest of the entire fluid:172
PV A =PV PVrest (7)
PVA is the part of the PV that is linked to the balanced dynamics and, at173
first order, it corresponds to the quasigeostrophic PV (Davis and Emanuel,174
1991; McIntyre and Norton, 2000; Herbette et al., 2003).175
The PV of the state at rest is given by the stratification at rest:176
PVrest =~
f.~
O⇢|⇢=fz@z⇢|⇢=fz
@⇢z(⇢)=fz
h(8)
An important point is that in Eq. 7 PVA has to be calculated along surfaces177
of constant density. This is underlined by the |⇢symbol in Eq. 8, which is178
valid for both non-hydrostatic and primitive equations. The stratification at179
rest ⇢is associated with the adiabatic rearrangement of the density to get a180
horizontally uniform field (Holliday and Mcintyre, 1981; Kang and Fringer,181
2010) and it is generally not easy to determine. PVA is thus often used in182
idealised configurations where the fluid is at rest in some area (generally at183
the edge of the domain see sections 5.2 and 5.3 below). Alternatively, PVA184
can be associated with small scale processes, superposed on a larger scale185
circulation. The reference state can then be approximately determined as186
a spatial average (over a distance that is much larger than the processes187
scales).188
9
3. Alternative expressions for PV189
3.1. Divergence form190
In the following, the calculations rely on general mathematical properties191
relating divergence, curl and gradient of 3D fields and integral properties of192
these operators, whose general forms are recalled in Appendix A.193
Previous studies have shown that Ertel PV, as defined in Eq. 2, can194
be expressed in divergence form (see Schneider et al., 2003; Vallis, 2006).195
Trivial manipulations (explained in Appendix A, see Eq. A.1) lead to the196
following equivalent expressions for the PV in divergence form (remember197
~
Ua=~
U+~
⌦⇥~ris the absolute velocity, see Fig. 1):198
PV
Ertel =div(~
Ua⇥~
O⇢) (9a)
=div(⇢(~
r⇥~
Ua)) (9b)
=div(~
U⇥~
O⇢)div(⇢~
f).(9c)
Notice that these expressions are exact, whatever the evolution (diabatic or199
adiabatic) of PV and have been reported and/or used before, in particular200
in atmospheric sciences (see Haynes and McIntyre, 1987; Bretherton and201
Schar, 1993; Schneider et al., 2003; Vallis, 2006). Here we demonstrate that202
they also lead to consistent and convenient practical approach to calculating203
and analysing PV in ocean modelling.204
3.2. Implication for the integral of PV205
Using Ostrogradsky-Stokes theorem (see Appendix A), the previous di-206
vergence form of the PV simplifies the calculation of the integral of PV
Ertel
207
10
over a volume V. It can be calculated from the knowledge of the den-208
sity, velocity or relative vorticity fields around the surface @Vcontaining V.209
Equations 9 give the exact expressions:210
ZZZ
V
PV
Ertel dV =ZZ
@V
⇢(~
r⇥~
Ua).d~
S(10a)
=ZZ
@V
(~
Ua⇥~
O⇢).d~
S(10b)
=ZZ
@V
⇢~
f.d~
SZZ
@V
(~
U⇥~
O⇢).d~
S. (10c)
The previous expressions follow from the definition of PV and do not depend211
on equations governing its evolution. They represent exact instantaneous212
diagnostics of net PV within a volume and should not be confused with213
the general flux form of the PV evolution equation (Haynes and McIntyre,214
1987).215
4. Applications to the calculation of PV216
In this section, we discuss how the divergence formulation, and its asso-217
ciated integral constraints Eq. 10, yield an easier way to diagnose PV and218
maintain balances between volume integral of PV and boundary conditions219
(Eq. 10).220
4.1. PV diagnostics for numerical models221
The diagnosis of PV from numerical model outputs is generally cum-222
bersome if the literal form (Eq. 2 or 5) is chosen as it implies numerous223
gradients calculated at di↵erent grid points, which then have to be aver-224
aged. The use of the divergence form simplifies the PV calculation and also225
preserves Eq. 10.226
11
As they are used in the majority of ocean circulation models, we consider227
a 3D C-grid, which are 3D extensions of the horizontal Arakawa C-grid (see228
Fig. 2 and Arakawa and Lamb, 1977). Using Cartesian coordinates, we229
start from the divergence form of PV (9b) rewritten as:230
PV
Ertel =div(⇢(~
⇣+~
f))
=@x(⇢(⇣x+fx)) @y(⇢(⇣y+fy)) @z(⇢(⇣z+fz)) (11)
where ~
⇣=~
r⇥~
Uand:231
⇣x=@yw@zv
⇣y=@xw+@zu
⇣z=@xv@yu. (12)
The elementary cell for which PV is calculated has the density values232
at its corners (see Fig. 2). As is clear from Fig. 2, ⇣zvalues need to233
be calculated at the center of lower and upper sides of the cell. It can be234
calculated using the circulation along edges of the cell lower and upper sides.235
An interesting property of 3D C-grid is that this is straightforward, thanks236
to the position of the velocity points (located at the middle of edges parallel237
to the velocity component). Density is averaged over the 4 density points238
located at the side corners. The same calculation is also valid for the other239
sides of the cell.240
As a result, the PV of the cell can easily be calculated from physical241
fields within this single cell. We get:242
12
Figure 2: Elementary cell, for a 3D C-grid, used for the calculation of PV. We consider
Cartesian coordinates (x, y, z) associated with indices (i, j, k).
13
⇣x
i,j,k =wi,j,k wi,j1,k
yvi,j,k vi,j,k1
z
⇣y
i,j,k =wi,j,k wi1,j,k
y+ui,j,k ui,j,k1
z
⇣z
i,j,k =vi,j,k vi1,j,k
xui,j,k ui,j1,k
y,(13)
and finally243
PV
i,j,k =⇢xi,j,k(⇣x
i,j,k +fx
i,j,k)⇢xi1,j,k (⇣x
i1,j,k +fx
i1,j,k)
x
⇢yi,j,k(⇣y
i,j,k +fy
i,j,k)⇢yi,j 1,k (⇣y
i,j1,k +fy
i,j1,k )
y
⇢zi,j,k(⇣z
i,j,k +fz
i,j,k)⇢zi,j,k1(⇣z
i,j,k1+fz
i,j,k1)
z,(14)
where244
⇢xi,j,k =1/4(⇢i,j,k +⇢i,j,k1+⇢i,j1,k +⇢i,j1,k1) (15)
is the density calculated at the position of ⇣x
i,j,k (see Fig. 2), and so forth for245
the other components. The Coriolis components fx/y/z
i,j,k are calculated at the246
location of the ⇣x/y/z
i,j,k points. Note that for the specific discretization of the247
3D C-grid (see Fig. 2), the divergence form leads to a compact expression248
of PV : in Eq. 14 PV is calculated using density and velocity values from a249
single grid cell.250
Equation 14 has a flux form, which ensures that, given a volume V,251
the integral of PV calculated over Vusing the accumulation of individual252
cells or using Eq. 10 exactly match, thus preserving the general balances253
between integral of PV and boundary conditions for any volume. Flux form254
14
PV expressions can be derived for B-grids or other grids, with a similar255
property.256
4.2. General PV expression in isopycnal coordinates257
The integral constraints 10 may be used for an easier derivation of the258
expression of PV in any coordinate systems and for the full Navier-Stokes259
equations. As an example, we calculate PV using the isopycnic coordinate260
⇢instead of the geopotential coordinate z(see section 4a of Schneider et al.,261
2003). This is of interest as the interpretation of the PV evolution, in262
particular the PV anomaly, has to be made along isopycnic surfaces (Hoskins263
et al., 1985).264
For the sake of simplicity, we just replace the vertical Cartesian coordi-265
nate zby ⇢and we keep the Cartesian (x, y) coordinates in the horizontal266
(see Fig. 3). Other systems (for instance spherical) can be used without267
much more complications. We also keep the orthogonal Cartesian elemen-268
tary vectors (
~
i,~
j,~
k) associated with axis (Ox, Oy, Oz) (see Fig. 3) to express269
all vectors.270
In this framework, z=z(x, y, ⇢) is the vertical position of isopycnic271
surfaces, and to calculate PV, we will use Eq. 10b, which only requires the272
evaluation of the density gradient ~
O⇢=@x⇢~
i+@y⇢~
j+@z⇢~
k,butusingthe273
(x, y, ⇢) coordinates. To do so, we use:274
h=@⇢z=1/@z⇢
@xz|y,⇢=h@
x⇢|y,z
@yz|x,⇢=h@
y⇢|x,z
The density gradient is then given by:275
15
~
O⇢=1
h[@xz~
i+@yz~
j~
k] (16)
Figure 3: Coordinate system (x, y, ⇢) and elementary volume and surfaces used to calculate
PV
Ertel using the isopycnic coordinate.
Equation 10b is then applied to an elementary volume bounded by two276
isopycnic surfaces sketched in Fig. 3:277
ZZZ
V
PV
Ertel dV =ZZ
@V
(~
Ua⇥~
O⇢).d~
S
=[(~
Ua⇥~
O⇢).d~
S]@V(17)
where [.]@Vis the flux through all surfaces delimiting V. Note that ~
Ua=278
ua~
i+va~
j+wa~
kremains the absolute velocity field expressed in the279
orthogonal Cartesian system.280
16
Since the flux across isopycnic surfaces (⇢±⇢) is null and since the other281
surfaces are simple (vertical planes of constant yor x), Eq. 17 gives:282
PV⇢
Ertel V=[( ~
Ua⇥~
O⇢).
~
i2y2h⇢]x+x
xx
[(~
Ua⇥~
O⇢).~
j2x2h⇢]y+y
yy(18)
Given that V=2x2y2z=2x2y2h⇢ and283
~
Ua⇥~
O⇢=1
h(va+wa@yz,uawa@xz, ua@yz+va@xz) (19)
Eq. 18 gives:284
PV⇢
Ertel =@x(va+wa@yz)|⇢@y(ua+wa@xz)|⇢
h
=@x(v+w@yz)|⇢@y(u+w@xz)|⇢+fzfy@yz
h(20)
which is a generalised form of Eq. 6 with additional terms (in particular all285
components of the Coriolis e↵ect). The terms (u+w@xz)|⇢,(v+w@yz)|⇢
286
represent the projection of the velocity field on the plane tangent to the287
isopycnic surface.288
This exact general result can also be derived using Eq. 2, with a change289
of coordinate. But the calculations based on Eq. 10 o↵er a straightforward290
method.291
4.3. Integration of PV in a ”layer”292
We consider a volume Vconstituted of a ”layer” embedded between293
two isopycnic surfaces associated with densities ⇢1and ⇢2, that can outcrop294
at the surface or bottom (see Fig. 4). The total PV contained within V295
17
may be deduced from Eq. 10c and trivial calculations (taking advantage296
of the fact that the boundaries @Vof the layer are partly delimited by297
isentropic/isopycnic surfaces, and some rearrangements using Eq. A.4).298
This leads to the following form, which depends only on physical fields299
at the surface and bottom outcropping regions:300
ZZZ
V
PV
Ertel dV =ZZ
Ss+Sb+Sw
(~
U⇥~
O⇢).d~
S
+[
ZZ
Ss
(⇢1⇢s)d~
S+ZZ
Sb+Sw
(⇢1⇢b)d~
S+ZZ
S⇢2
(⇢1⇢2)d~
S].~
f
(21)
where ⇢s(x, y) is the density at the ocean surface and ⇢b(x, y)thedensity301
along the bottom of the ocean. This form takes advantage of the expression302
Eq. 10c to deal with volumes delimited by the two isopycnal surfaces S⇢1and303
S⇢2. Part of the layer boundaries are however associated with outcropping304
surfaces where density varies (Ss,Swand Sbsee Fig. 4). The first right305
hand side term of Eq. 21 depends on ~
U⇥~
O⇢and has to be evaluated along306
these surfaces. For this term, depending on the boundary condition used,307
it may be more convenient to switch back to a form in ⇢~
⇣like in Eq. 10a.308
This has to be done carefully using Eq. A.4 (see Appendix A). For instance309
we obtain for the surface Ss:310
ZZ
Ss
(~
U⇥~
O⇢).d~
S=ZZ
Ss
(⇢1⇢s)⇣sdxdy (22)
Finally, notice that the bottom surface has been divided in ”Sidewalls” and311
”Bottom” regions (Swand Sb, see Fig. 4), possibly associated with di↵erent312
boundary conditions. This is artificial if both surfaces are associated with313
18
Figure 4: General shape of a layer, bounded by two isopycnic surfaces S⇢1and S⇢2,de-
termining a volume where we integrate PV. Outcropping may occur at the surface (Ss)
and at the bottom (Sb). As sketched in the upper plot (a) ”Sidewalls” (Sw) and ”Bot-
tom” (Sb) surfaces are sometimes distinguished in numerical model. In this case, layers
outcropping at the surface and sidewalls can have special constraints (b), as discussed in
section 4.4.
19
the seafloor but we did make a di↵erence for the sake of generality. For314
instance in academic configurations, such as a rectangular basin, boundary315
conditions at the walls and at the bottom can di↵er.316
4.4. Impermeability theorem317
The impermeability theorem (Haynes and McIntyre, 1987, 1990) states318
that there is no net transport of PV across isopycnic (or isentropic) surfaces,319
whatever the evolution. As already shown by Vallis (2006), Eq. 10b is a320
straightforward demonstration of this theorem. Indeed, across such surfaces,321
d~
Sis parallel to ~
O⇢and Eq. 10b shows that they do not contribute to the322
calculation of the PV volume integral, whatever the evolution of the isopy-323
cnic surfaces. Thus, if there are no outcropping regions and the isopycnic324
surfaces are closed, the volume integral of Ertel PV within closed isopycnic325
surfaces is and remains null, whatever the evolution. Alternatively, modi-326
fication of the volume integral of PV in an isopycnic layer is only possible327
when isopycnic surfaces outcrop (Haynes and McIntyre, 1987).328
This principle can be slightly extended. Considering a layer without329
surface outcropping, and considering a no-slip boundary condition at the330
ocean bottom ( ~
Uw=~
Ub=~
0), Eq. 21 gives:331
ZZZ
V
PV
Ertel dV =[ ZZ
Sb+Sw
(⇢1⇢b)d~
S
+ZZ
S⇢2
(⇢1⇢2)d~
S].~
f. (23)
If ~
Uw=~
Ub=~
0, the density distribution along the bottom can only be332
modified by diabatic (mixing) e↵ects along the bottom. If the latter are333
negligible, the density field along the bottom is constant, and Eq. 23 then334
shows that there is no modification of the volume integral of PV. Indeed, in335
20
this case, both terms in the right hand side of Eq. 23 are constant. This336
is obvious for the first term. The second term is simply the scalar product337
of ~
f(constant) and the net S⇢2surface vector. The latter only depends on338
the position of the edge of the surface, defined by the ⇢2contour along the339
bottom, and thus constant too (an alternative way to demonstrate this is to340
transform the second term using Eq. A.4, see Appendix A). To conclude,341
with no-slip boundary conditions, the volume integral of PV is only modified342
if there exists mixing of the density near the bottom. In practice, the free-343
slip boundary condition is often preferred in ocean circulation models, the344
implication for the generation of PV will be discussed below (section 5.4).345
Another case of interest is when outcropping only occurs at the surface346
and sidewalls (Fig. 4 b). In numerical models, sidewalls are sometimes347
considered vertical and the fycomponent of the Coriolis vector is also ne-348
glected, so that ~
f.d~
S= 0. If no-slip boundary conditions are used, many349
terms disappear in Eq. 21 and we then obtain:350
ZZZ
V
PV
Ertel dV =ZZ
Ss
(~
U⇥~
O⇢)d~
S+ZZ
Ss
fz(⇢1⇢s)dS (24)
This draws attention to the potential importance of sloping boundaries and351
the fycomponent for the volume integral of PV at basin scale. It also352
shows that the surface terms in Eq. 24 are of special interest and we further353
evaluate their contributions in the next section.354
5. Applications to specific balances355
As discussed in the introduction, there exists a strong link between ocean356
circulation and the PV field, from mesoscale eddies to large scale currents.357
Equation 21 shows that there exists a balance between a volume integral358
21
of PV and boundary conditions. Using the divergence form of PV and the359
local PV calculation discussed in section 4.1 allows to preserve this balance.360
This is important for the physical interpretation of model outputs in terms361
of PV.362
In this section, we illustrate how the balance can be used at several scales363
and for various processes in realistic or idealised configurations, for which364
some terms in Eq. 21 can be easily evaluated from observations (e.g. the365
surface ones), simplified or neglected (e.g. for no slip boundary conditions).366
In section 5.1 we discuss how time variations of large scale volume inte-367
gral of PV can be related to surface fields for both models and observations.368
At mesoscale, surface density anomalies play a role similar to PVA369
(Bretherton, 1966). In sections 5.2 and 5.3 we show how Eq. 21 can be370
applied to isolated vortices and jets. We show that the balance leads to a371
precise relationship between surface density anomalies and PVA integrals,372
which has to be satisfied for isolated vortices and jets.373
Finally, in section 5.4 we show how Eq. 21 can be applied to study the374
modification of PV in the bottom boundary layer, underlining the strong375
impact of the boundary conditions (free/no-slip).376
5.1. Surface outcropping regions as indicators of the circulation of deep lay-377
ers378
For some choices of boundary conditions Eq. 21 reduces to Eq. 24. In379
addition, PV can be quickly modified by diabatic processes at the surface380
(Thomas, 2005; Morel et al., 2006; Thomas and Ferrari, 2008; Thomas et al.,381
2013; Wenegrat et al., 2018). We can thus hypothesise that the surface term:382
22
Isurf =ZZ
Ss
[(⇢1⇢s)~
f(~
U⇥~
O⇢s)].d~
S
dominates the time evolutions of the integral of PV within a deeper layer,383
which is itself linked to modification of the circulation (Rhines and Young,384
1982a,b; Luyten et al., 1983; Holland et al., 1984; Rhines, 1986; Thomas and385
Rhines, 2002; Polton and Marshall, 2003; Deremble et al., 2014). Comparing386
Isurf from numerical models and observations is thus of interest.387
Using d~
S=~
k dxdy (where ~
kis the vertical elementary vector), Isurf can388
be rewritten:389
Isurf =ZZ
Ss
[(⇢1⇢s)~
f(~
U⇥~
O⇢s)].~
k dxdy (25)
Note that the integral in Eq. 25 only requires the knowledge of surface390
fields, in particular (~
U⇥~
O⇢s).~
konly depends on the horizontal gradient of391
the surface density. Isurf can be calculated directly for numerical models.392
For observations, satellite observations (possibly complemented by in situ393
surface drifter observations) provide good estimates of the surface circulation394
over most of the ocean down to scales of order 25 km (see for instance Sudre395
and Morrow, 2008; Abernathey and Marshall, 2013; Rio et al., 2014). To do396
so, the surface current is split into a geostrophic component and a component397
induced by the wind stress:398
~
Us=~
Ugeo +~
U⌧(26)
The geostrophic component ~
Ugeo and the associated relative vorticity can399
be calculated from the knowledge of the sea surface height (SSH) observed400
by satellite altimetry:401
23
~
Ugeo =g
fz
~
k⇥~
OSSH (27)
The wind induced surface current can be evaluated from satellite scat-402
terometer observations and using the wind induced Ekman spiral which gives403
(see Cushman-Roisin and Beckers, 2011):404
~
U⌧=~⌧ ⇡/4
w
pfz⌫(28)
where ⌫is the turbulent eddy viscosity and405
~⌧ ⇡/4
w=⇢a
⇢o
CDkWk~
W⇡/4(29)
where ~
W⇡/4is the surface wind but whose orientation has been rotated by406
⇡/4, ⇢a/⇢ois the ratio of the air to ocean density and CD'3.103is the407
turbulent transfer parameter. As a result, the surface term contributing to408
the calculation of the observed PV within a layer (Eq. 25) can be written:409
Isurf =ZZ
Ss
(⇢1⇢s)fz[( g
fz
~
k⇥~
OSSH +⇢aCDkWk~
W⇡/4
⇢opfz⌫)⇥~
O⇢s].~
k dxdy
(30)
and can be calculated from the observed sea surface density (calculated using410
SSS and SST from SMOS, Aquarius and microwave satellite observations),411
SSH and surface wind (all fields generally available over most of the ocean at412
1/4oresolution). We believe the comparison of Isurf from numerical models413
(Eq. 25) and from observations (Eq. 30) can provide an interesting new414
diagnostic for the validation of global or basin scale numerical models.415
24
5.2. Constraints for coherent isolated vortices416
Most observed eddies in the ocean are isolated3(Chelton et al., 2011).417
In QG and SQG models, for coherent isolated vortices, the volume integral418
of PVA and surface density field are linked (Morel and McWilliams, 1997;419
Assassi et al., 2016). We here extend this balance to Ertel PVA.420
Consider a flat earth for which ~
f=(0,0,f
z) (f-plane approximation)421
and an axisymmetric vortex over a flat bottom (see Fig. 5 b-d). For the422
sake of simplicity, we also hypothesise that ⇢is constant at the bottom and423
that PVrest is spatially uniform (linear stratification at rest).424
Integrating the PVA over the control volume Vo(see Fig. 5 b-d) gives:425
ZZZ
Vo
PVA dV =ZZZ
Vo
(PV
Ertel PV rest)dV
=ZZZ
Vo
(~
f.~
O⇢PVrest)dV
+ZZZ
Vo(~
r⇥~
U).~
O⇢dV (31)
By using that PVrest =fz(⇢1
s⇢b)/H, Eq. A.1b and the fact that the426
vortex is isolated, we get:427
ZZZ
Vo
(~
f.~
O⇢PVrest)dV =ZZ
Ss
(⇢s⇢1
s)fzdx dy (32)
and428
3An isolated vortex has a velocity field that decreases more rapidly than 1/r, where r
is the distance from its center, and the horizontal integral of its vorticity is null at any
level.
25
Figure 5: Vertical density structures for axisymmetric vortices having negative (b) and
positive (d) surface anomalies. Vo(dashed contour) is the volume of integration and ris
the distance form the vortex center. The background stratification at rest is indicated in
panels a and c.
26
ZZZ
Vo(~
r⇥~
U).~
O⇢dV =ZZ
@Vo
⇢(~
r⇥~
U).d~
S
=ZZ
Ss
⇢s⇣surf dx dy +O(1
r)
=ZZ
Ss
(⇢s⇢1
s)⇣surf dx dy +O(1
r) (33)
where ⇣surf =@xv@yuis the relative vorticity at the surface, ⇢1
sis the429
surface density at rest or the surface density far from the vortex center, and430
(⇢s⇢1
s) is the surface density anomaly associated with the vortex4.431
Integration of Eq. 31 over the whole (infinite) domain shows that PV432
anomalies associated with isolated vortices have to satisfy:433
ZZZPVA dV +ZZ
Ss
(⇢s⇢1
s)(⇣surf +fz)dx dy = 0 (34)
This extends the integral constraints found in Assassi et al. (2016), which is434
modified for strong surface vorticity (when |⇣surf |'fz)5. This is the case435
for submesoscale vortices (Lapeyre et al., 2006; Klein et al., 2008; Capet436
et al., 2008; Roullet et al., 2012; Gula et al., 2015; Molemaker et al., 2015;437
Capet et al., 2016).438
4In Eq. 33, the last line is obtained since RR
Ss⇣dxdy= 0 for isolated vortices. The
O(1/r) term accounts for the integration over the bottom and lateral boundaries (dashed
contours in Fig. 5). In particular, the lateral contribution scales as |⇢H2⇡r@
zU(r)|
O(1/r). The O(1/r) rate of decrease is symbolic and the term simply indicates that these
contributions vanish when r! 1 .
5Strictly speaking, strong anticyclonic vortices, for which ⇣surf <fz, could even
reverse the sign of the deep PVA, but these structures are subject to inertial instability
and are not long lived structures.
27
Thus, for isolated vortices, a positive surface density anomaly is accom-439
panied with negative PVA. A positive surface density is equivalent to a440
positive Dirac delta sheet of PVA (Bretherton, 1966). A similar constraint441
holds for a negative density anomaly. Hence, the generalised PVA structure442
of isolated vortices has both positive and negative values, which implies op-443
posite sign PV gradient and opens the possibility of barotropic/baroclinic444
instabilities (Charney and Stern, 1962; Ripa, 1991). This has an impact445
on the evolution (stability and displacement) of the vortex (see Morel and446
McWilliams, 1997). In idealised studies dealing with the dynamics of iso-447
lated vortices, instability of the initial vortex structure can spoil the analysis448
and it is preferential to use specific methods, based on the inversion of stable449
PV structures, to initialise isolated vortices in models (see Herbette et al.,450
2003).451
Moreover, the constraint Eq. 34 can have implications for methodologies452
deriving velocity fields of vortices from surface density observations. The453
methodologies empirically generate PVA distributions based on large-scale454
PV distributions or statistical correlations between surface density obser-455
vations and PVA (Lapeyre et al., 2006; Lapeyre and Klein, 2006; Lapeyre,456
2009; Ponte et al., 2013; Wang et al., 2013; Fresnay et al., 2018). In general,457
the derived PVA distributions do not satisfy constraint 34. The consequence458
is that the velocity field of a reconstructed vortex decreases slowly, which459
can lead to spurious calculations near lateral boundaries (the methodolo-460
gies often consider periodic boundary conditions). It could be interesting461
to modify the methodologies so as to satisfy Eq. 34 in the vicinity of each462
vortex. We however have no clue on the spatial distribution of the PVA463
from the constraint (PVA poles, crown, vertically aligned or not, vertical464
position within the water column, possibly multiple poles of opposite sign,465
28
...) and the reconstruction of the vertical vortex PVA have thus to be done466
carefully.467
5.3. Constraints for jets and surface fronts468
Similar constraints can be found for density fronts associated with jet-469
like currents. We consider a 2D configuration with no variation in the y470
direction. In 2D, Eq. 10 becomes471
ZZ
S
PV
Ertel dS =Z@S
⇢(~
r⇥~
Ua).~n dl
=Z@S
(~
Ua⇥~
O⇢).~n dl
=Z@S
⇢~
f.~n dl Z@S
(~
U⇥~
O⇢).~n dl (35)
Consider a 2D front outcropping at the surface but with a constant density472
along a flat bottom (see Fig. 6). The velocity field can be written ~
U=473
V(x, z)~
j,whereVis the velocity component along the yaxis. For jet-like474
currents the velocity vanishes away from the front: V(x! ±1,z) = 0.475
The stratification is di↵erent on both sides of the front and varies from476
¯⇢1(z)to¯⇢+1(z).477
For this configuration, the determination of the reference PV, associated478
with the state at rest, is slightly more delicate, as we hypothesised that both479
the left and right edges of the front are at rest. It has however to be chosen at480
the left edge as only this side covers the entire density range. The reference481
PV is thus PV1
rest and we then integrate PVA from x=1 to x=L.482
Again, for the sake of simplification, we hypothesise that ~
f=(0,0,f
z) and483
PV1
rest is spatially uniform. Trivial manipulations yield an equation similar484
to Eq. 33:485
29
Figure 6: Vertical density structures for a surface outcropping front. S(dashed contour)
is the surface of integration from x=1 to x=L.
30
ZZ
S
PVA dS =ZZ
S
PV
Ertel PV 1
rest dS
=Zx=L
1
(⇢⇢1)|z=0 (⇣z+fz)|z=0 dx
+Zz=0
z=H
(@z⇢V)|x=Ldz (36)
Assuming the velocity has a jet-like structure, V(x=L, z) becomes small486
enough so that the last term in Eq. 36, can be neglected. Given the density487
structure discussed here (see Fig. 6), (⇢⇢1)|z=0 is positive, which488
shows that a negative PVA must exist below the outcropping region for jets489
(if (⇣z+fz) remains positive). Opposite sign generalised PVA is necessarily490
associated with opposite sign PVA gradients and to instability (Charney491
and Stern, 1962). Similarly to isolated vortices, integral constraint 36 can492
be useful to study the instability of surface fronts and for methods aiming at493
reconstructing the ocean at mesoscale and submesoscale via an estimation494
of PVA within the water column (Lapeyre et al., 2006; Ponte et al., 2013;495
Spall, 1995; Boss et al., 1996; Manucharyan and Timmermans, 2013).496
5.4. PV modification by bottom boundary layer processes497
To study the modification of PV by -necessarily- diabatic processes, Eq.498
4 complemented with the knowledge of diabatic terms is needed (Benthuy-499
sen and Thomas, 2012; Molemaker et al., 2015; Gula et al., 2015, 2019).500
However, as shown next, integral constraints may provide an interesting501
way to monitor the PV evolution within an isopycnic layer intersecting the502
topography.503
To do so let us consider the development of a bottom boundary layer in504
2D, with no variation in the ydirection (Fig. 7).505
31
Figure 7: Vertical density structures in the deep ocean, near a topography. We consider a
2D configuration and we follow the evolution of a layer determined by two isopycnic levels
⇢1and ⇢2intersecting the topography. The initial velocity profile and the positions of the
isopycnic levels (a) are modified by some diabatic processes (b).
32
We also consider that there is no outcropping at the surface and we fol-506
low a control area A2Dbounded by two isopycnic surfaces ⇢1and ⇢2,the507
topography and a vertical boundary located at a distance L1sufficiently508
large so that we can consider being away from the boundary layer and un-509
a↵ected by the diabatic processes (the stratification and velocity field are510
unchanged, see Fig. 7). Integration of PV over this area gives (see Eq. 35):511
ZZ
A2D
PV
Ertel dA =[ZSw
(⇢1⇢b)~n d l +ZS⇢2
(⇢1⇢2)~n d l ].~
f
ZSw
(~
U⇥~
O⇢).~n dl ZS1V1@z⇢1dz (37)
Given its definition, the last term in Eq. 37 does not vary.512
The isopycnic levels initially intersect the topography at x= 0 and513
x=L, and along the topography the velocity field is Vo~
j(Fig. 7a). After514
some diabatic processes, involving the viscous boundary layer and diapycnal515
mixing, the velocity profile and the position of isopycnic surfaces are modi-516
fied. The positions of the intersection with the topography are now x=L1
517
and x=L2and the velocity field along the topography is V~
j(Fig. 7b).518
Some trivial manipulations give:519
ZSw
(⇢1⇢b)~n . ~
fdl=fzZL2
L1
(⇢1⇢b)dx
ZS⇢2
(⇢1⇢2)~n . ~
fdl=fz(⇢2⇢1)(L1L2)
ZSw
(~
U⇥~
O⇢).~n dl =Z⇢2
⇢1Vd⇢(38)
Assuming a linear variation of the density along the bottom topography,520
this gives for the initial condition (see Fig. 7):521
33
ZZ
A2D
PV
Ertel dA =fz(⇢2⇢1)(L1L
2)Vo(⇢2⇢1)ZS1V1@z⇢1dz
(39)
and after the diabatic modification:522
ZZ
A0
2D
PV
Ertel dA =fz(⇢2⇢1)(L1L1+L2
2)
V(⇢2⇢1)ZS1V1@z⇢1dz (40)
where Vis the mean velocity along the bottom topography (where the av-523
erage is weighted by density). The net modification of the volume integral524
of PV within the layer is thus:525
ZZ
layer
PV =(⇢2⇢1)fzX⇢1/⇢2
bot (⇢2⇢1)V⇢1/⇢2
bot (41)
where V⇢1/⇢2
bot =VVois the modification of the mean velocity field along526
the bottom and within the layer ⇢1/⇢2, and X⇢1/⇢2
bot =(L1+L2)L
2is the527
modification of the mean xposition of the layer along the bottom.528
If no-slip conditions are chosen at the bottom, we recover that only529
density mixing along the bottom can modify the volume integral of PV530
within a layer, as already discussed in section 4.4. The time evolution of the531
volume integral of PV then only depends on the variation of the position532
of the intersection of the isopycnic layer: it is negative if the layer goes533
downslope (destratification case as illustrated in Fig. 7) and positive if534
the layer goes upslope (restratification case). Our results are qualitatively535
consistent with Benthuysen and Thomas (2012), despite the fact that we536
consider a layer and not a fixed box for the volume integral of PV.537
34
Equation 41 allows the possibility to consider free-slip bottom condi-538
tions. Free-slip boundary conditions is the constraint usually used in nu-539
merical models and can provide an additional modification of the volume540
integral of PV if viscous e↵ects are considered, as first imagined by D’Asaro541
(1988). These viscous e↵ects have to be added to the e↵ect of the modifi-542
cation of density studied in Benthuysen and Thomas (2012) and discussed543
above. Equation 41 shows that they superimpose when calculating the vol-544
ume integral of PV and generally act similarly. Since our results are only545
diagnostics, we have to ”imagine” the evolution of the velocity and density546
fields along the boundary to evaluate the possible PV modification. If we547
consider a velocity field with the shallow region on its right (Vo<0, as de-548
picted in Fig. 7), in the northern hemisphere, the bottom friction develops549
a downslope Ekman flux that leads to destratification and mixing induces a550
negative volume integral of PV variation. We can also assume that bottom551
friction also acts so as to reduce the strength of the velocity along the bot-552
tom topography, so that |V|<| Vo|. This leads to V⇢1/⇢2
bot >0 and again to553
a negative volume integral of PV variation. Similarly an initial current with554
shallow region on its left would lead to a positive variation. This is consis-555
tent with recent high resolution numerical results, using free-slip boundary556
conditions (see Molemaker et al., 2015; Gula et al., 2015; Vic et al., 2015;557
Gula et al., 2016, 2019) .558
However, as discussed above, the important dynamical quantity is not559
necessarily the volume integral of PV. The key quantity is the PVA within560
an isopycnic layer. We can diagnose the mean PVA evolution within the561
boundary layer by dividing the volume integral of PV by the volume of562
the followed fluid (or its area A2Dand A0
2Din 2D, see Fig. 7). When563
all isopycnic surfaces remain parallel, this volume is constant (as is the564
35
case in Benthuysen and Thomas, 2012, for instance), the mean PVA is565
similar to the volume integral of PV and all previous results thus apply566
to the mean PVA. However, when this is not the case, the modification of567
PVA is more complex and also involves PV dilution or concentration within568
a layer which respectively gains or loses mass (see Haynes and McIntyre,569
1990; Morel and McWilliams, 2001). This process is e↵ective whenever570
there exists variation of turbulence along the topography, which is the case571
if the bottom slope or the velocity field vary spatially. In addition, global572
mass conservation requires that the depletion of one layer coincides with573
the inflation of another layer. Thus, di↵erential diapycnal mixing in bottom574
boundary layers is probably ubiquitous in realistic configurations and we575
can expect the creation of both positive and negative PV anomalies.576
6. Summary and discussion577
6.1. Summary578
In the present paper, we have used three di↵erent formulations of Ertel579
PV in divergence form (see Schneider et al., 2003, and Eq. 9) to calculate a580
volume integral of PV from the knowledge of physical fields at the surface581
encompassing the volume. The divergence form and associated integral con-582
straints have then been used to enable easier calculation of PV for numerical583
models, also preserving the balances between boundary conditions and PV.584
This has been explored in more details for specific physical processes at585
di↵erent scales.586
We have also shown that the integral constraints associated with the587
divergence form lead to an easier calculation of the PV expression for non588
Cartesian coordinate systems. We have in particular illustrated this by589
36
calculating its expression in isopycnal coordinates for the general Navier-590
Stokes equations.591
We have then considered the volume integral of PV within a ”layer”592
delimited by two isopycnic surfaces and their intersections with the ocean593
surface and bottom. A general integral constraint was derived which allows594
to extend the PV impermeability theorem to no-slip conditions provided595
there is no density mixing along the topography. The integral constraint is596
then applied to several specific processes.597
We first explored the link between volume integral of PV and surface598
fields at basin scale and we proposed an indicator to evaluate the time599
evolution of the volume integral of PV within a layer provided it outcrops600
at the sea surface (section 5.1). We proposed an indicator Isurf , depending601
on physical fields at the surface, as the signature of deeper PV. The indicator602
can be easily calculated for models and compared to observations (it depends603
on physical fields that can be estimated using satellite observations: wind,604
sea surface height, surface temperature and salinity).605
When applied to isolated vortices or jets, given the equivalence between606
outcropping and surface PVA concentration (Bretherton, 1966), the balances607
indicate that such structures have opposite sign generalised PVA and are608
thus potentially unstable. It also provides a useful constraint to estimate609
PVA structures from surface information as currently attempted empirically610
(Lapeyre et al., 2006; Lapeyre and Klein, 2006; Lapeyre, 2009; Ponte et al.,611
2013; Wang et al., 2013; Fresnay et al., 2018).612
We finally applied the integral constraints to the modification of PV613
by diabatic processes within the bottom boundary layer. This provides a614
diagnostic of the PV evolution within a layer based on the displacement of its615
mean position and on the modification of the mean along slope velocity along616
37
the topography. It shows in particular that free-slip boundary conditions617
have potentially stronger e↵ects on the formation of PVA in the viscous618
boundary layer. Di↵erential mixing (variation of the density mixing along619
the topography) also leads to additional and possibly opposite sign PVA620
along the topography.621
6.2. Discussion622
Concerning the calculation of PV in numerical models, the divergence623
form approach can be adapted to any type of grid (including unstructured624
grids). In numerical models, the main problem is however Lagrangian con-625
servation of PV during the (adiabatic) evolution of the flow. This principally626
relies on numerical schemes used in the model. There exists debates on the627
optimality of numerical grids (for instance between the Charney-Phillips grid628
and the 3D C-grid, see Arakawa and Moorthi, 1988; Bell, 2003) but a fair629
comparison relies on comparable numerical schemes too: numerical schemes630
have to be optimised for the conservation of PV for each grid (see Winther631
et al., 2007). When this is established, the influence of the PV diagnostic632
on the conservation property is interesting to assess too, even though this633
influence is expected to be marginal compared to numerical schemes.634
Concerning the Isurf indicator, we hypothesised that the time evolution635
of the integral of PV in a layer was mostly induced by the evolution of636
the surface fields. Recent studies (Ferrari et al., 2016; McDougall and Fer-637
rari, 2017; de Lavergne et al., 2017; Callies and Ferrari, 2018) have however638
shown that mixing is bottom intensified at large scale and that it is as-639
sociated with strong upwelling/downwelling circulations along the bottom640
topography which control the abyssal circulation overturning. According to641
what is discussed here in section 5.4, this can also modify the average PV.642
38
The signature of the modification of the deep PV on surface and bottom643
boundary terms of the PV balance (Eq. 21) can be tested using numerical644
models (Deremble et al., 2014). Equation 14 can be used to calculate PV645
consistently with Eq. 21.646
An interesting perspective is to combine the present results with the wa-647
ter mass transformation (WMT) approach (Walin, 1982; Tziperman, 1986;648
Speer and Tziperman, 1992). If the surface contribution to the volume in-649
tegral of PV can be exactly estimated for numerical models, we have to rely650
on geostrophic and Ekman currents for observations, so that we may miss651
some important ageostrophic contributions to the surface current, in partic-652
ular associated with mixing. The WMT theory allows one to estimate the653
surface drift associated with mixing and heat fluxes and correct the surface654
observations where needed. The importance of this term for the PV balance655
can be assessed in models and the WMT approach provides a way to take656
this e↵ect into account in observations.657
Concerning the dynamics of isolated vortices and jets, the balances can658
be easily extended to take into account variations of density along the bot-659
tom (variations of bottom density have then to be included in Eq. 34 and660
36) and a variable stratification at rest (see Eq. B.6 in Appendix B). This661
implies that the PVA evaluation is also possibly influenced by the bottom662
conditions, so that it may be difficult to reconstruct PVA profiles from the663
knowledge of surface density anomalies alone. Our calculations used the664
f-plane approximation. On the -plane, weak vortices are dispersed into665
Rossby waves and their initial isolated nature can be rapidly lost. The re-666
sults we derive here are thus of interest mainly for coherent vortices whose667
PV structures is comprised of closed PV contours. For these vortices, we668
can neglect the variation of the Coriolis parameter and Rossby waves.669
39
Concerning modification of PV in the bottom boundary layer, the net670
modification of PV is also a function of time (Benthuysen and Thomas,671
2013): the velocity and stratification in the bottom boundary layer do not672
reach instantaneously their equilibrium value (Benthuysen and Thomas,673
2012). Thus, the final modification of PV along a boundary depends on674
the time a fluid parcel will remain in contact with the boundary layer. A675
Lagrangian perspective shows that 3D e↵ects are important for realistic676
conditions: when a circulation encounters a bottom boundary, a fluid parcel677
will be in contact with the boundary layer for a limited time period which678
is a function of the boundary and circulation shapes (see Fig. 8). Both679
frictional e↵ects and diapycnal mixing will modify the PV value of the fluid680
parcel and the strength of the created PVA which eventually separates from681
the boundary.682
The identified processes for PV modification in the bottom boundary683
layer have physical grounds but their implementation in numerical simula-684
tions is a delicate issue as the result also depends on the choices of several685
parameters (turbulent viscosity and di↵usion, but also numerical schemes,686
boundary conditions and closure schemes for momentum and tracers in the687
bottom boundary layer). Further studies are needed to evaluate the respec-688
tive strength of each process in numerical simulations and in nature. The689
present results give exact diagnostics that can be helpful for that purpose.690
40
Figure 8: Schematic view of the modification of the PV of a fluid parcel that enters and
exits a bottom boundary layer. The PV modification is a function of the time period
the parcel remains within the bottom boundary layer, which is itself a function of the
circulation and topography characteristics.
41
Appendix A. General mathematical properties691
For the sake of application to PV, we name ~
U,~
⇣and ⇢the fields used692
in the following equations, but the latter are exact general mathematical693
results whatever the meaning of the ~
U,~
⇣and ⇢fields.694
First let us recall some basic properties for the divergence and curl of695
arbitrary fields:696
div(~
U⇥~
B)=(
~
r⇥~
U).~
B(~
r⇥~
B).~
U, (A.1a)
div(⇢~
⇣)=~
⇣.~
O⇢+⇢div(~
⇣),(A.1b)
~
r⇥(⇢~
U)=⇢(~
r⇥~
U)~
U⇥~
O⇢,(A.1c)
div(~
r⇥~
U)=0,(A.1d)
~
r⇥(~
O⇢)=~
0.(A.1e)
697
Using ~
U=~
Uaand ~
B=~
O⇢in A.1a, and ~
⇣=~
r⇥~
Uain A.1b, Eq. A.1698
allow to derive the divergence forms of the PV (Eq. 9).699
We also use the Ostrogradsky-Stokes theorems for the integration of700
divergence and curl fields:701
ZZZ
V
div(~
A)dV =ZZ
@V
~
A.d~
S(A.2)
and
ZZ
S
(~
r⇥~
A).d~
S=Z@S
~
A.d~
l(A.3)
where Vis a finite volume, @Vis its external surface and d~
Sis an elementary702
surface oriented outward and is perpendicular to @⌦,Sis a surface, @Sis703
42
its boundary and d~
lis an elementary line oriented parallel to @Sand in the704
trigonometric direction when Sis ”seen from above” (see Fig. A.9).705
Figure A.9: Vector directions for the calculation of volume to surface to line integrals
(Stokes theorem).
Finally, Eq. A.1 and A.3 also give:706
ZZ
S
⇢(~
r⇥~
U).d~
S=ZZ
S
(~
U⇥~
O⇢).d~
S+Z@S
⇢~
U.d
~
l. (A.4)
All these integral properties allow the derivation of Eq. 10 and its alter-707
native forms.708
Appendix B. Generalised constraints in nonuniform stratification709
Appendix B.1. Generalised PV710
The definition of PV (Eq. 2) could be changed and ⇢can be replaced711
by G(⇢)whereGrepresents a general function. The generalised PV form is712
43
thus:713
PV
Ertelgen =(~
r⇥~
U+~
f).~
OG(⇢)
=G0(⇢)PV
Ertel (B.1)
and such a change does not alter the basic properties associated with PV714
and discussed in the paper.715
The integral of the generalised Ertel PV satisfies all results discussed716
above. In particular, Eq. 10 becomes:717
ZZZ
V
PV
Ertelgen dV =ZZ
@V
G(⇢)(
~
r⇥~
Ua).d~
S
=ZZ
@V
(~
Ua⇥~
OG(⇢)).d~
S
=ZZ
@V
G(⇢)~
f.d~
SZZ
@V
(~
U⇥~
OG(⇢)).d~
S
(B.2)
The integration within a layer (Eq. 21) gives:718
ZZZ
V
PV
Ertelgen dV =[
ZZ
Ss
(G(⇢1)G(⇢s)) d~
S
+ZZ
Sb+Sw
(G(⇢1)G(⇢b)) d~
S
+ZZ
S⇢2
(G(⇢1)G(⇢2)) d~
S].~
f
ZZ
Ss+Sb+Sw
(~
U⇥~
OG(⇢)).d~
S(B.3)
Appendix B.2. Potential Vorticity Anomaly719
For a fluid at rest, where the velocity field and vorticity are null and the720
stratification only depends on the vertical coordinate, the previous gener-721
alised form gives:722
44
PVrest
Ertelgen =G0(⇢)fz@z⇢
=fz@z[G(⇢(z)) ] (B.4)
where fzis the local vertical component of the Coriolis vector and ⇢(z)is723
the reference profile of the stratification at rest. Choosing G(X)=¯⇢1(X),724
where ¯⇢1is the inverse of the function ¯⇢(z) (so that G(⇢(z)) = z), yields725
PVrest
Ertelgen =fz: the reference PV is spatially uniform (f-plane approxi-726
mation).727
Using the generalised form of PV given in Eq. B.1 and B.4, we calculate728
the generalised PVA:729
PVA
gen =PV
Ertelgen PVrest
Ertelgen
=(~
r⇥~
U+~
f).~
OG(⇢)+fz(B.5)
Since the stratification at rest is constant, the calculation performed in730
section 5.2 can be reproduced to lead to the general integral constraints for731
isolated vortices in a nonuniform stratification:732
ZZZPVA
gen dV +ZZ
Ss
(G(⇢)G(⇢1
s))(⇣+fz)dx dy =0
(B.6)
Note that G=¯⇢1is a monotonically increasing function, so that all733
the physics discussed in section 5.2 remains qualitatively valid.734
Acknowledgements735
Yves Morel is supported by the program ”IDEX attractivity chairs”736
from Universit´e de Toulouse (TEASAO project) and CNES (french space737
45
agency; project TOSCA/OSTST ”Alti-ETAO”). This work also benefited738
from the Copernicus Marine Environment Monitoring Service (CMEMS)739
DIMUP project. CMEMS is implemented by Mercator Ocean in the frame-740
work of a delegation agreement with the European Union. J. Gula benefited741
support from LEFE/IMAGO through the Project AO2017-994457-RADII.742
A. Ponte benefited support from CNES for his participation to the SWOT743
Science Team (project ”New Dynamical Tools”). The authors acknowledge744
discussions with Prof. Peter Haynes (Chair holder of the TEASAO project)745
and Drs. Leif Thomas, Jef Polton and John Taylor which helped improving746
this manuscript. This work has been drastically improved thanks to the crit-747
icisms, comments and careful reading of anonymous reviewers and of Ocean748
Modelling editors. Their encouragements have also been a strong support749
for us and their suggestions led to a far better presentation of our results.750
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