Numerical instabilities of vector invariant momentum equations on rectangular C-grids
Michael J. Bell
a∗and Pedro S. Peixoto
bcand John Thuburn
baMet Office, Fitzroy Rd, Exeter, UK
bCollege of Engineering, Mathematics and Physical Sciences, University of Exeter, UK
cInstituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Brazil
∗Correspondence to: Met Office, Fitzroy Rd, Exeter, EX1 3PB.
The linear stability of a stably stratified Boussinesq fluid on an f-plane with a constant velocity field is examined for dynamics governed by the vector invariant hydrostatic primitive equations discretised using both the original and modified forms of the Arakawa- Lamb (AL) scheme and Sadourny’s energy and enstrophy conserving (een) scheme. In height and isopycnal coordinates the problem can be solved by separation of variables using vertical normal modes and two slightly different discretisations of the linearised shallow water equations (SWEs). These linearised SWEs determine the stability of the same constant velocity field in shallow water when the slope of the bottom makes the fluid of constant depth. Simple analytical expressions are derived for the smallest equivalent depths obtained using Charney-Phillips and Lorenz grids. Expressions for the growth rates of the instabilities are then obtained for all Fourier modes for both the AL and een schemes, in their original and modified forms, for both height and isopycnal coordinates. The matrix determining the stability of the modified schemes is shown to be Hermitian so they are stable for all linearised disturbances. All perturbations in isopycnal models are also shown to be neutrally stable even in the original schemes. Test cases are proposed for assessing the stability of new numerical schemes using the SWE.
Key Words: Hollingsworth instabilities, vector invariant momentum equations, shallow water equations, separation of variables, energy and enstrophy conservation
Received . . . Citation:. . .
1. Introduction
The vector invariant form of the momentum equations expresses the advection of momentum as the sum of the gradient of the kinetic energy and the vector product of the velocity and the vorticity. This form is used to derive Kelvin’s circulation theorem (Pedlosky 1987) and was used by Sadourny (1975) to discretise the shallow water equations (SWEs). In that paper Sadourny proposed two schemes, demonstrating that one conserves energy and the other, now known as the ens scheme, conserves the volume integral of potential enstrophy (the square of the potential vorticity). He later devised the een (energy and enstrophy) scheme, which for the SWEs conserves the total energy for general flows and enstrophy for non-divergent flows.
This scheme was tested in a hydrostatic primitive equation (HPE) model using pressure based sigma coordinates by Burridge and Haseler (1977). Arakawa and Lamb (1981), hereafter referred to as AL, derived the Arakawa-Lamb (AL) scheme which, for the SWEs, conserves the total energy and enstrophy for general flows. AL also provided a proof of the conservation properties of the een scheme.
Hollingsworthet al. (1983), HKRB hereafter, reported that the een scheme as implemented by Burridge and Haseler (1977) was prone to near-grid- scale instabilities which reduced the kinetic energy of the jet stream particularly in their higher resolution model.
They provided an heuristic linear stability analysis which accounted for the 3-grid-point structure of the instabilities in the horizontal and predicted that the growth rates should be proportional tof u/cwheref is the Coriolis parameter, uis the speed of the flow in the basic state andc=√
gHis the gravity wave speed. This prediction was consistent with their experimental results for different speedsuand their finding that the modes with highest vertical wavenumber (and smallest speeds c) are the most unstable ones. From the dispersion relationship derived by HKRB it is clear that the instability is a form of destabilised inertia-gravity wave. HKRB also proposed a modified scheme involving
reformulations of the kinetic energy gradient and the fluxes at the faces of velocity cells for the een scheme and showed that it was effective in suppressing the instabilities. A similarly modified form of the AL scheme can also be used (see section 6 of AL).
There has recently been renewed interest in these instabilities for two reasons. Firstly some ocean models now use grids which resolve the Rossby radius of deformation very well. Ducousso and le Sommer (2015) found that the NEMO model, which uses the een scheme, when configured with a 1 Nautical Mile grid spacing had significantly reduced kinetic energy in the mesoscale flow unless the kinetic energy was reformulated as proposed by HKRB. Secondly a number of researchers such as Niˇckovi´cet al. (2002), Thuburn (2008), Ringleret al.
(2010),Skamarocket al.(2012) andGassmann(2013) are seeking to develop atmospheric models using meshes with triangular, hexagonal or pentagonal elements employing the vector invariant momentum equations.Skamarocket al.
(2012) note that their scheme is prone to the Hollingsworth instability and choose a formulation of the kinetic energy which appears to suppress it. Appendix B of Gassmann (2013) discusses the Hollingsworth instability from an historical perspective and proposes a method for choosing the formulation of the kinetic energy on regular hexagonal grids so as to minimise the size of the term in the momentum equations which lead to the instabilities.
As noted above HKRB provided a good initial theoretical analysis of the instability, but their derivation of the dispersion relationship for the instability included the neglect of a term which was only justified by a rather ad hoc argument. Also some aspects of the occurrence of the instability have not been clarified since the work of HRKB.Arakawa(2000) notes that the family of consistent energy and enstrophy conserving schemes (including the een and AL schemes) that AL derived generally behave well for the SWEs and that the Hollingsworth instabilities arise in pressure or sigma coordinates “at least in part”
because of the formal application of the schemes in these
coordinates in which the layer depth his replaced by the thickness of model layers despite the fact that the model levels are not material surfaces. This has left developers of new dynamical cores uncertain how to test their schemes using the SWEs. This is very inconvenient for them and a better understanding of the occurrence of the instabilities in easily accessible variants of the SWEs is highly desirable.
This paper has two main aims. The first is to show that the stability of the een and AL schemes for an idealised 3D basic state is determined by separable solutions that are the product of vertically varying normal modes and solutions to linearised SWEs. The resulting linearised SWEs are also shown to determine the stability of the flow in shallow water in a container whose bottom slopes at just the right angle for the depth of the fluid to be constant in the presence of the flow. This should allow the potential for Hollingsworth instabilities in 3D problems to be explored with new numerical schemes using (properly specified) 2D problems.
The second aim is to derive the matrices determining the dispersion relationships for these linearised SWEs and to analyse them. It is shown that the modifications proposed by HKRB and AL to the een and AL schemes recover the Hermitian form of the stability matrix and hence make the schemes stable for any linear disturbance to the idealised basic states. The original een and AL schemes in isopycnal models are also shown to be neutrally stable to all perturbations. Numerical results and an expression for the instabilities in height coordinates suggest that the most unstable perturbations are fairly closely aligned with the grid.
The idealised 3D basic state whose stability is examined consists of a uniform horizontal flow(u1, v1)(independent ofx,yandz) in a stably stratified fluid on anf-plane. Here u1 andv1 are constant eastward and northward velocities respectively,xandyare eastward and northward increasing coordinates respectively, andzis height. It is more natural to consider this configuration in an oceanic context, where variations in the surface height of the ocean can easily occur
and affect the pressures at all depths, than in an atmospheric context. For this reason the analysis is presented using the Boussinesq equations which are appropriate for the ocean (rather than the equations of state for a perfect gas appropriate for the atmosphere).
The linear stability analysis of the states described in this paper is most safely approached by writing out the full non-linear governing equations in discretised form and the description of the basic state, then deriving from these the linearised equations and then finally deriving the separable solutions. For the equations discretised in height coordinates this approach is unnecessarily lengthy and it is easier (and safe) to linearise and derive the separable solutions using the continuous equations and then discretise. Section2performs this calculation. Section 3 starts from the fully non-linear SWEs with sloping bathymetry and notes that in continuous form they reduce to the linearised SWEs derived in section2. It then presents a succinct formulation of the discretisation of the SWEs for the een and AL schemes appropriate both for height coordinate models and as originally formulated by AL.
It concludes by writing down the linearised equations for these formulations. Section 4 considers the idealised 3D basic state for a model formulated in isopycnal coordinates, derives the form of linearisation the een and AL schemes would use and peforms the analysis of the separation of variables. This analysis for isopycnal coordinates has been postponed until after section 3 because some aspects of the discretisation need to be taken into account when the equations are linearised.
Section 5 derives the vertical discretisation of the modes in isopycnal and height coordinates. The vertical modes with the highest vertical wavenumbers have small equivalent depths as one would expect from the vertical modes for the continuous problem. It is shown that on the Lorenz grid the smallest equivalent depths reduce as the number of vertical levels, K, increases at a rate which is a factor ofK2faster than that of the continous modes. This
result is related to the presence of the computational mode on the Lorenz grid.
Section6 starts by reducing the linearised equations for the een schemes to eigenvalue problems involving3 by 3 matrices. The matrix is written in a non-dimensional form and it is shown that the growth rate (non-dimensionalised by f0) depends on six non-dimensional quantities and that for the modified form of the een scheme the matrix is Hermitian and has real eigenvalues and hence that the scheme is stable. The same conclusions are shown to hold for the AL scheme. Section 7 analyses the instabilities obtained with the original schemes in height coordinates, section 8 summarises the main characteristics of the growth rates obtained by numerical determination of the eigenvalues and section9illustrates the nature of these instabilities obtained by integration of the SWEs. The latter also proposes test cases for doubly-periodic Cartesian and spherical domains that could be used to test whether new numerical schemes suffer from these instabilities. Finally section10shows that all linear perturbations are neutrally stable for the original schemes in isopycnal coordinates and section11provides a concluding summary.
The symbols used in this paper are listed in TablesItoIII.
The vertical component of the relative and total vorticities will be denoted byζandZrespectively with
ζ= ∂v
∂x−∂u
∂y, Z =f0+ζ. (1) The Coriolis parameter f will be taken to be a constant value f0, ρ00 will denote a constant density, and the horizontal kinetic energy per unit mass will be denoted by
Ke=1
2(u2+v2). (2) For simplicity the equations will be written in Cartesian coordinates and discretised on a C-grid with uniform, but not necessarily isotropic, grid spacing. The arrangement of variables on the C-grid is illustrated in Figure1.
Figure 1.The staggering of variables on the C-grid. This figure and the indexing is based onArakawa and Lamb(1981)
The following difference and average operators will be used. Letξdenote any of the coordinate directionsx,yand z,ιdenote its discrete indexing andψdenote any function ofξ. Then
ψξι ≡1
2(ψι−1/2+ψι+1/2), (∆ξ)ι≡ξι+1/2−ξι−1/2, (δξψ)ι≡ ψι+1/2−ψι−1/2
(∆ξ)ι
, (διψ)ι≡(∆ξ)ι(δξψ)ι. (3)
The index for which the quantity is calculated is usually suppressed. All of these operators commute with each other and obey the associative laws of arithmetic.Adcroftet al.
(1997) provide a useful summary of identities they satisfy.
2. Separation of variables in height coordinates
2.1. Governing equations
For simplicity the governing equations in height coordinates will be taken to be a form of the hydrostatic Boussinesq equations suitable for a liquid written in the Cartesian coordinates introduced earlier. For this case the Bernoulli functionΦis given by
Φ =p/ρ00+Ke (4)
where p is the pressure field. The horizontal momentum equations in vector invariant form are then
∂u
∂t −Zv+w∂u
∂z =−∂Φ
∂x +AmDmu, (5)
Table I. Table of Greek symbols for primary variables. The third column is the first equation where the term appears
Greek - Lower case
αtoδ linear combinations ofqorZ (200)
δι a difference operator (3)
δI,δB Kronecker delta functions (64) ǫ,φ contributions to Zu and Zv in
the AL scheme
(57) ζ vertical component of the rela-
tive vorticity
(1) η height of the free surface (22) ι the grid index in any direction (3) κ wavenumber of perturbation in
x-direction
(124) λ wavenumber of perturbation in
y-direction
(124) µE,µA coefficients related to averaging
inx
(41) νE,νA coefficients related to averaging
iny
(41) ξ a coordinate in any direction (3) π standard definition for circle
̟ Doppler shifted non-
dimensional frequency
(156), (186)
ρ density (4)
σ inverse of stratification in isopycnal model
(77) φ a contribution to Zu in the AL
scheme
(57)
ψ any function ofξ (3)
χ averaging coefficients (157)
ξ any coordinate direction (3)
ω non-dimensional frequency of perturbation
(124) Greek - Upper case
∆ the difference between neigh- bouring grid points
(3) Λ a non-dimensional parameter
for vertical modes
(111)
Φ the Bernoulli function (4)
∂v
∂t +Zu+w∂v
∂z =−∂Φ
∂y +AmDmv. (6) where w is the vertical velocity and Am is a coefficient of viscosity and Dm is a diffusive operator such as the horizontal Laplacian. The viscous terms will be set to zero except in this section and section 7 when the stabilising effect of viscous terms on perturbations with small equivalent depths are discussed. The fluid will be taken to be in hydrostatic balance,
∂p
∂z =−ρg, (7)
whereρis the density andgis gravity, and incompressible
∂u
∂x+∂v
∂y +∂w
∂z = 0. (8)
Table II. Table of Roman symbols for primary variables. The third column is the first equation where the term appears
Roman - Lower case
a1, a2, a3 coefficients (172)
a a parameter (161)
b height of the bottom (in SWEs) (34)
c speedc2=gH
cκ,cλ cosines of wavenumbers (125)
f the Coriolis parameter (1)
g gravity (7)
h function ofzrelated toworz (26) m a constant related to the equiva-
lent depth
(108)
p pressure (4)
q potential vorticity (40)
sκ,sλ sines of wavenumbers (125)
t time (5)
u x-component of horizontal velocity
(1) v y-component of horizontal
velocity
(1) w vertical component of velocity (5) x east-west coordinate, increases
eastward
(1)
y north-south coordinate,
increases northward
(1) z vertical coordinate, increases
upward
(5) Roman - Upper case
Am coefficient of viscosity (withm subscript)
(5) Aij matrix element for AL scheme (147)
C a complex constant (114)
C1 a contribution (201)
DM determinant (184)
D total derivative (as inD/Dt) (10) Eij matrix element for een scheme (134)
Fu, Fv Froude numbers (130)
G a geometric product (202)
H the total depth (or equivalent depth)
H an Hermitian matrix (142)
I the identity matrix (142)
J a non-dimensional quantity (160) Ke horizontal kinetic energy (2) K the number of vertical levels (102)
M Montgomery potential (78)
N−1 the number of zeros in the vertical
(117)
P, Q coefficients (188)
Rc twice the ratio of the Rossby radius and grid spacing
(131)
Ru,Rv Rossby numbers (132)
S a non-dimensional quantity (176) Tu,Tv modified Rossby numbers (182)
X grid aspect ratio (131)
W a constant in normal mode soln (114) Z vertical component of the total
vorticity
(1)
As described in section 1.6.2 ofVallis(2006) the potential density, ρe will be taken to be conserved following the motion
Dρe
Dt = 0, ρe=ρ+ρ00z Hρ
. (9)
Table III. Table of subscripts and superscripts. The third column is the first equation where the term appears
Subscripts
ρ compression as inHρ (9)
e equivalent inHeand potential inρe
i imaginary part (161)
i thex-coordinate index Fig1 j they-coordinate index Fig1
k the vertical index (100)
r real part (161)
s sound as incs (9)
x thex-coordinate (5)
y they-coordinate (6)
A Arakawa and Lamb (AL) scheme (5)
B modified AL scheme (145)
E een scheme (41)
F modified een scheme (42)
H horizontal as innablaH (5)
H upper boundary as inbH (83)
I isopycnal (64)
M matrix (184)
00 constant density (4)
0 the stably stratified component of the basic state
(12) 1 the constant velocity component of the
basic state
(13) Superscripts
µ averaging inx (41)
ν averaging iny (41)
′ a perturbation (15)
∗ transports through faces of cells (40) φˆ a function of the vertical coordinate (22)
φ˜ a function ofx,yandt (22)
φ an average ofφ
Here
D Dt = ∂
∂t+u∂
∂x +v ∂
∂y +w∂
∂z, (10) is the material derivative following the motion andHρis the scale height for compression given in terms of the speed of soundcsbygHρ=c2s.
The configuration will be taken to be unbounded inxand yand to have a flat boundary atz=−Hwhere the vertical velocity is zero. Attention will be focussed solely on the baroclinic modes for which to a very good approximation the upper boundary atz= 0also has zero normal velocity so
w= 0, z= 0,−H. (11) The barotropic mode satisfies the shallow water equations (to a very good approximation) and is not considered further in this section.
2.2. Assumed basic state
The assumed basic state consists of a stably stratified density field ρ0(z)that is in hydrostatic balance with the pressure fieldp0(z)
dp0
dz =−ρ0g, (12) and a horizontal velocity field with componentsu=u1and v=v1which do not depend onx,y,z ort. This velocity field is in geostrophic balance with a pressure fieldp1which is independent ofz,
p1=f ρ00(v1x−u1y). (13)
The other fields are all equal to zero or constants:
w0= 0, Z0=f0. (14)
2.3. Perturbed equations
The evolution of very small amplitude perturbations can be determined by linearising the above equations about the basic state. Denoting the perturbations by primed quantities and neglecting products of perturbations the horizontal momentum equations for the perturbations are given by
∂u′
∂t −f0v′−ζ′v1=−∂Φ′
∂x +AmDmu′, (15)
∂v′
∂t +f0u′+ζ′u1=−∂Φ′
∂y +AmDmv′. (16) where
ζ′=∂v′
∂x −∂u′
∂y, Φ′= p′ ρ00
+ (u1u′+v1v′). (17)
Hydrostatic balance for the perturbations is
∂p′
∂z =−ρ′g, (18)
the incompressibility condition is
∂u′
∂x +∂v′
∂y +∂w′
∂z = 0, (19)
and the density perturbations satisfy
∂ρ′
∂t +u1
∂ρ′
∂x +v1
∂ρ′
∂y +w′dρe
dz = 0. (20)
Finally the boundary conditions are
w′ = 0, z= 0,−H0. (21)
In summary the above set has five unknowns u′, v′, w′, ρ′ and p′ that are constrained by five equations and the boundary conditions (21).
2.4. Separable solutions
Comparing the five equations just summarised with those in section 6.11 of Gill (1982) one sees that they enjoy separable solutions of the same form. Departing slightly from the order of Gill’s derivation let
u′= p(z)ˆ gρ00
˜
u(x, y, t), v′ = p(z)ˆ gρ00
˜
v(x, y, t), p′= ˆp(z)˜η(x, y, t).
(22)
Then firstly
Φ′= p(z)ˆ gρ00
Φ(x, y, t),˜ Φ(x, y, t) =˜ gη˜+u1u˜+v1v,˜ (23) and secondly the horizontal momentum equations are satisfied if
∂u˜
∂t −f0v˜−ζv˜ 1=−∂Φ˜
∂x +AmDmu,˜ (24)
∂˜v
∂t +f0u˜+ ˜ζu1=−∂Φ˜
∂y +AmDmv.˜ (25) Following Gill we let
ρ′= ˆρ(z)˜η, w′ = ˆh(z) ˜w(x, y, t). (26)
Substituting (22c) and (26a) into (18) we obtain dˆp
dz =−gρ.ˆ (27)
Substituting (26) into (20) and introducing the separation constantHewe obtain
Hew˜= ∂η˜
∂t +u1
∂˜η
∂y+v1
∂η˜
∂y, (28)
Heρ(z) =ˆ −dρe
dz ˆh(z). (29) Substituting (22) and (26b) into (19) we can set
ˆ p gρ00
= dˆh
dz, (30)
∂u˜
∂x +∂˜v
∂y
+ ˜w= 0. (31) Eliminatingw˜from (31) using (28) we obtain
∂η˜
∂t +u1
∂η˜
∂x+v1
∂η˜
∂y +He
∂˜u
∂x+∂˜v
∂y
= 0. (32)
The horizontal- and time-varying functions (denoted by tilde superscripts) are then governed by the shallow water equations (24), (25) and (32). The vertical structure functionsˆh(z),ρ(z)ˆ andp(z)ˆ are determined by (27) (29) and (30) and the boundary conditions obtained from (21) and (26)
ˆh= 0, z= 0,−H. (33)
3. Linearisation of the shallow water equations
3.1. Governing equations
Consider a layer of shallow water of constant density on an f-plane in which the bottom of the fluid is at heightband the depth of the fluid layer isη. Then the Bernoulli potential is given by
Φ =g(η+b) +Ke, (34)
and the momentum and continuity equations are
∂u
∂t −Zv=−∂Φ
∂x, (35)
∂v
∂t +Zu=−∂Φ
∂y. (36)
∂η
∂t + ∂
∂x(ηu) + ∂
∂y(ηv) = 0. (37) 3.2. Assumed basic state
The assumed basic state has a constant horizontal velocity field (u1, v1) which is again in geostrophic balance with the surface pressure field η+b. For the given horizontal velocity field the bathymetry is chosen to be given by
gb=gb0+f0(v1x−u1y), (38)
where b0 is a constant, so that the depth of the fluid, denoted byH, is independent of position. Linearisation of the continuous equations (35) - (37) about this basic state easily yields equations isomorphic to (24), (25) and (32).
The numerical implementation of the SWEs reported in section7uses periodic boundary conditions in bothxand y. For this case the zonal flow is balanced by adding a force termFxto the rhs of (35) which is equal to−f0v1/H.
3.3. Discretisation used by the een scheme
The discretisation on a C-grid of ζ as defined by (1a) is given by
ζ=δxv−δyu (39) and the discretisation of the x- and y-derivatives of the Bernoulli function is given byδxΦandδyΦrespectively.
The een and AL schemes for shallow water models re- writeZvin (35) asqv∗andZuin (36) asqu∗where
q= Z
η(q), u∗=η(u)u, v∗=η(v)v, (40) and the(q),(u)and(v)superscripts indicate values that are centred at theq,uandvpoints. It is shown in section4that
the separable solutions for isopycnal coordinates follow this form. For height coordinates the separable solutions useZv rather thanqv∗in (5) and hence should useZvrather than qv∗in (35) andZurather thanqu∗in (36).
The averaging operators needed to define the modified een scheme are given by
ψµE≡ 1 3ψ+2
3ψxx, ψνE≡ 1 3ψ+2
3ψyy, (41) where theE subscript indicates the expression is relevant to the een scheme. Additional averaging operatorsµF and νF which allow the original and modified schemes to be concisely defined are given by
ψµF ≡ψµE, ψνF ≡ψνEfor the modified scheme, ψµF ≡ψ, ψνF ≡ψ for the original scheme.
(42)
For shallow water and isopycnal coordinate models the een scheme calculates (40) using
η(q)=ηxy, η(v)=ηyµF, η(u)=ηxνF. (43)
This averaging of the layers at velocity points is used for both height and isopycnal coordinates in the discretisation of the continuity equation (37)
∂η
∂t +δx(ηxνFu) +δy(ηyµFv) = 0. (44) This is required to ensure conservation of total (kinetic plus potential) energy with the modified form of the kinetic energy ((50) below). In height coordinate models or the SWEs calculated usingZvandZurather thanqv∗andqu∗ (as described above) the calculations do not use (40a) or (43a).
The een scheme for the SWEs and isopycnal coordinates then discretises the termZv=qv∗in (35) using
(Zv)E= 2
3(qxv∗)xy+2
3 qyv∗xy
−1
3 qv∗xy , (45)
and the termZuin (36) using
(Zu)E =2
3(qyu∗)xy+2
3 qxu∗xy
−1
3 qu∗yx . (46)
Expressions (45) and (46) are noted in HKRB. A brief derivation of them from the expression for the een scheme used by AL is presented in Appendix A. For height coordinates and the related SWEs
(Zv)E =2
3(Zxv)xy+2 3
Zyvxy
−1
3(Zvx)y, (47) and the termZuin (36) using
(Zu)E=2
3(Zyu)xy+2 3
Zxuxy
−1
3(Zuy)x. (48) The original version of the een scheme takesKe to be given by
2KE=u2x+v2y (49) and the modified version sets
2KE=u2xνE+v2yµE. (50)
3.4. Discretisation used by the AL scheme
As for the een scheme the original and modified AL schemes can be concisely written by defining the averaging operators
ψµA≡ψxx, ψνA ≡ψyy, (51) where theA subscript indicates the expression is relevant to the AL scheme, and the associated “modified” averaging operators
ψµB ≡ψµA, ψνB ≡ψνA for modified scheme, ψµB ≡ψ, ψνB ≡ψ for original scheme.
(52)
Similarly to (43) the AL scheme for isopycnal coordinate models sets
η(q)=ηxy, η(v)=ηyµB, η(u)=ηxνB, (53)
and discretises the continuity equation as
∂η
∂t +δx(ηxνBu) +δy(ηyµBv) = 0. (54)
It then discretises the termZv=qv∗in (35) using
(Zv)A=(qxyv∗y)x+ 1
48δi[(δiδjq) (δjv∗)]
+δi ǫu∗x
+ǫδiu∗x,
(55)
and the termZuin (36) using
(Zu)A=(qxyu∗x)y+ 1
48δj[(δiδjq) (δiu∗)]
+δj φv∗y
+φδjv∗y,
(56)
in which (3d) definesδiandδjand
ǫ= 1
12δjqx, φ= 1
12δiqy. (57) Ketefian and Jacobson(2009) note that the AL scheme can be expressed in the form given by (55) - (56) and these equations are derived in detail in the appendices ofKetefian (2006). For height coordinates the same expressions apply withqreplaced byZ,u∗byuandv∗byv.
The original version of the AL scheme takes Ke to be given by (49). AL propose a modified form for the kinetic energy in their equation (6.1). A form which is more similar to that used above for the een scheme whose first term gives the same gradients ofKAas the form proposed by AL is given by
2KA=u2xνA+v2yµA− 1
12δiδj(uyyvxx). (58) The last term in (58) has been introduced here to cancel contributions arising from the Coriolis terms proportional toǫandφin (55) and (56).
3.5. Linearised equations for the een scheme
Let
u=u1+u′, v=v1+v′, η=H+η′, (59)
whereu′,v′andη′are small perturbations andH has been used to denote the undisturbed depth of the water.
Using δxψ2= 2ψxδxψ one finds that the linearised kinetic energy gradient in thex- andy-directions for (50) is given by
δxKE′ =u1δxu′xνF +v1δxv′yµF, (60)
δyKE′ =u1δyu′xνF +v1δyv′yµF. (61)
Linearising (45) and (46) and (47) and (48) one obtains
(Zv)′E=f0v′xy+ζ′yµEv1+δI
f0v1
H
η′µF −η′µExyy , (62) (Zu)′E=f0u′xy+ζ′xνEu1+δI
f0u1
H
η′νF −η′νExxy , (63) where
δI =
0 for height coordinates, 1 for isopycnal coordinates.
(64)
The final term in each of (62) and (63) is zero for the original scheme in height coordinates and zero for the modified scheme for both height and isopycnal coordinates.
Using (39) in (60) and (62), one sees that the discrete linearised form of (35) is given by
∂u′
∂t −f0v′xy+gδxη′+u1δxu′xνF +v1δyu′yµE+ v1δx v′yµF −v′yµE
−δI
f0v1
H η′µF −η′µExyy
= 0. (65)
The first line above consists of terms corresponding to those present in the continuous equations. The second line consists of additional terms arising from the discretisation employed which have the potential to give rise to spurious effects. Similarly using (39) in (61) and (63) the discrete
linearised form of (36) is
∂v′
∂t +f0u′xy+gδyη′+u1δxv′xνE+v1δyv′yµF+ u1δy u′xνF −u′xνE
+δIf0u1
H η′νF −η′νExxy
= 0, (66)
in which the first line again contains terms corresponding to those in the continuous equations and the second line contains additional terms. Linearising (44) one also finds that
∂η′
∂t +u1δxη′xνF +v1δyη′yµF +H(δxu′+δyv′) = 0.
(67)
3.6. Linearised equations for the AL scheme
Defining perturbations as in (59) again one finds that for both the original and modified schemes the kinetic energy gradients can be expressed as
δxKA′ =u1δxu′xνB+v1δxv′yµB
−δB
u1
12δiδj
u1δxv′µA+v1δxu′νA ,
(68)
δyKA′ =u1δyu′xνB+v1δyv′yµB
−δB
1 12δiδj
u1δyv′µA+v1δyu′νA .
(69)
where
δB=
0 for the original scheme, 1 for the modified scheme.
(70)
Linearising (55) and (56) for isopycnal models and the corresponding expression for height coordinate models one obtains
(Zv)′A=f0v′xy+ζ′yµAv1−u1
12δiδjζ′µA +δIf0v1
H
η′µB−η′µAxyy ,
(71)
(Zu)′A=f0u′xy+ζ′xνAu1−v1
12δiδjζ′νA +δIf0u1
H
η′νB−η′νAxxy ,
(72)
whereδIis defined in (64).
Using (68) and (71) one sees that the discrete linearised form of (35) is given by
∂u′
∂t −f0v′xy+gδxη′+u1δxu′xνB+v1δyu′yµA
− 1 12
u1δyδiδju′µA+δBv1δxδiδju′νA +v1δx
v′yµB−v′yµA
+(1−δB)
12 u1δxδiδjv′µA
−δI
f0v1
H
η′µB−η′µAxyy
= 0.
(73)
As in (65) the first line of (73) consists of terms corresponding to those in the continuous equations and the other lines consist of potentially troublesome terms associated with the discretisation. Similarly using (69) and (72) the discrete linearised form of (36) is
∂v′
∂t +f0u′xy+gδyη′+u1δxv′xνA+v1δyv′yµB
− 1 12
δBu1δyδiδjv′µA+v1δxδiδjv′νA +u1δy
u′xνB−u′xνA
+(1−δB)
12 v1δyδiδju′νA +δI
f0u1
H
η′νB−η′νAxxy
= 0.
(74)
Linearising (54) one obtains
∂η′
∂t +u1δxη′xνB+v1δyη′yµB+H(δxu′+δyv′) = 0.
(75)
4. Separation of variables in isopycnal coordinates
4.1. Governing equations
As in section 2 only a hydrostatic, adiabatic, Boussinesq fluid will be considered. Following section 3.9.1 ofVallis (2006) let
ρ=ρ00+δρ, p=p0(z) +δp, dp0/dz=−gρ00, (76) whereρ00is still a constant. The buoyancy and an inverse measure of the stratification for this system are then given by
b=−gδρ ρ00
, σ= ∂z
∂b. (77)
If the isopycnal coordinates were used in a layer model, σ would represent the thickness of the layers. The Montgomery potential and Bernoulli function are given by
M = δp
ρ00 −bz, Φ =M +K. (78)
The horizontal momentum equations are then
∂u
∂t −Zv=−∂Φ
∂x, (79)
∂v
∂t +Zu=−∂Φ
∂y, (80)
in which the partial derivatives are evaluated with b held constant and the diapycnal velocities have been set to zero.
The hydrostatic equation takes the form
∂M
∂b =−z, (81)
and for an ideal Boussinesq fluid the continuity equation is given by
Dσ Dt = ∂σ
∂t +u∂σ
∂x +v∂σ
∂y =−σ ∂u
∂x+∂v
∂y
, (82)
in which all partial derivatives are again evaluated with b held constant. Denoting the buoyancy atz= 0andz=−H byb(z= 0) =b(0)andb(z=−H) =b(−H)respectively, the boundary conditions of no normal flow are given by
Dz
Dt = 0, b=b(0), b(−H). (83)
4.2. Assumed basic state
The assumed basic state is the same as that in the previous section. The stably stratified state is expressed as a profile z0(b)that is in hydrostatic balance
dM0
db =−z0. (84)
The horizontal velocity field(u1, v1)is again in geostrophic balance with a pressure fieldp1so,
M =M0(b) +M1, M1=f(v1x−u1y). (85)
The relative vorticity is again zero and the total vorticity, Z0=f0, is independent of position.
4.3. Linearised equations
The horizontal momentum equations for the perturbations for the case of the een scheme are given by
∂u′
∂t −f0v′−ζ′v1−f0v1
σ0
σ′µE−σ′µF
=−∂Φ′
∂x , (86)
∂v′
∂t +f0u′+ζ′u1+f0u1
σ0
σ′νE−σ′νF
=−∂Φ′
∂y . (87) where the additional terms involving variations inσwhich arise from discretisingZvfor example as(Z/σ)(σv)have been included in sufficient detail for their discrete form to be inferred later in the calculation and
ζ′ =∂v′
∂x −∂u′
∂y, Φ′=M′+ (u1u′+v1v′). (88) From (81), hydrostatic balance for the perturbations is given by
∂M′
∂b =−z′. (89) Continuity of mass, (82), gives
∂σ′
∂t +u1∂σ′
∂x +v1∂σ′
∂y =−σ0
∂u′
∂x +∂v′
∂y
, (90)
and the boundary conditions, (83), become
∂z′
∂t +u1
∂z′
∂x +v1
∂z′
∂y = 0, b=b(0), b(−H). (91) In summary the above set has four unknowns u′,v′, M′ andz′that are constrained by four equations and the above boundary conditions.
4.4. Separable solutions
Let
u′ = ˆM(b)˜u(x, y, t), v′ = ˆM(b)˜v(x, y, t), M′ = ˆM(b)g˜η(x, y, t).
(92)
Then
Φ′= ˆM(b) ˜Φ(x, y, t), Φ(x, y, t) =˜ gη˜+u1u˜+v1v,˜ (93) and the terms other than those involving σ in the horizontal momentum equations reduce to (24) and (25).
The additional terms are considered below Letting
z′= ˆh(b)˜η, (94) and substituting (92c) and (94) into (89) gives
gd ˆM
db =−ˆh. (95) Substituting (92) and (94) also into (90) and introducing the separation constantHeone obtains
∂η˜
∂t +u1∂η˜
∂x+v1∂η˜
∂y =−He
∂u˜
∂x +∂v˜
∂y
(96)
and
He
dˆh
db =σ0M .ˆ (97) The vertical structure of the additional terms in the horizontal momentum equation can now be considered.
Using (77) and (97) one sees that their vertical structure is given by
σ′ σ0
= η˜ σ0
dˆh db =η˜Mˆ
He
(98) Hence these additional terms have the same vertical structure as the other terms in the momentum equations and they do make the expected contributions to the momentum equations in the shallow water equations.
Clearly (96) is the same as (32), so the horizontal- and time-varying functions,u,˜ v˜andη˜are governed by the same
shallow water equations ((24), (25) and (32)) as before, though instead of (47) the discretisation of the Coriolis term in (24) for the een scheme uses (43) and (45). The vertical structure functionsMˆ(b)andz(b)ˆ are determined by (95) and (97) and the boundary conditions obtained from (91) and (94) namely
ˆh= 0, b=b(0), b(−H). (99)
5. Discretisations in the vertical
The natural discretisation in the vertical of the isopycnal model is to take the horizontal boundaries to lie at half levels and storeu,vandM at full levels andzat half levels as in Figure2b. (92) then implies thatMˆ is stored at full levels and (94) implies thatˆhis stored at half levels. Denoting the levels with a subscriptk, the level number increasing with height, (95) is discretised as
g( ˆMk+1−Mˆk) =−ˆhk+1/2(∆b)k+1/2. (100)
and (97) as
He(ˆhk+1/2−ˆhk−1/2) =σ0kMˆk(∆b)k. (101)
For a grid withKlevels, the boundary conditions (99) are simply
hˆ1/2= ˆhK+1/2= 0. (102)
The natural discretisation in the vertical of the level model is less clear cut and as is well known there are a number of options (Tokioka 1978;Thuburn and Woollings 2005). Most ocean models use the Lorenz grid illustrated in Figure2a in whichu,v,ρandpare stored on full levels andw is stored at half-levels. Thenpˆandρˆare stored at full levels andˆhat half-levels and (27), (29) and (30) are discretised respectively by
ˆ
pk+1−pˆk=−g
2(ˆρk+1+ ˆρk) (∆z)k+1/2, (103)
Figure 2.The arrangement of variables using (a) height coordinates and (b) isopycnal coordinates. In both gridsuandvare held at full levels, and the upper and lower boundaries are at half-levels.
Heρˆk=−dρe
dz 1 2
ˆhk+1/2+ ˆhk−1/2
, (104)
ˆ
pk(∆z)k=gρ00
ˆhk+1/2−ˆhk−1/2
. (105)
The boundary conditions (33) are discretised by
ˆh1/2= ˆhK+1/2= 0. (106)
One sees that (105) and (106) correspond to (101) and (102) and that (103) and (104) when combined correspond to (100) but in a form that involves more vertical averaging. The discretisation used above for the isopycnal model corresponds to that for the best category in Thuburn and Woollings (2005) obtained using a Charney- Phillips grid with potential temperature evaluated at half levels .
The impact of the vertical discretisation on the equivalent depth, He, which is the separation constant and the eigenvalue for the normal modes in the vertical can be illustrated for the case of uniform stratification and grid- spacing. Then (100) and (101) for the isopycnal coordinate model reduce to
ˆhk+3/2−2ˆhk+1/2+ ˆhk−1/2+m2ˆhk+1/2= 0, (107)
m2= σ0(∆b)2 gHe
= ∆z∆b gHe
. (108)
The last identity above follows from (77b) which implies that σ0= ∆z/∆b. Similarly (103) - (105) for the height coordinate model reduce to
hˆk+3/2−2ˆhk+1/2+ ˆhk−1/2+ n2
4
ˆhk+3/2+ 2ˆhk+1/2+ ˆhk−1/2
= 0, (109)
where
n2=− g ρ00
dρe
dz
∆z2 gHe ≈ ∆b
∆z
∆z2 gHe
=m2. (110)
Both (107) and (109) can be written in the form
ˆhk+3/2−2Λˆhk+1/2+ ˆhk−1/2= 0. (111)
For the isopycnal model one finds that
Λ = 1−m2/2, (112)
whilst for the height coordinate model
Λ = 4−n2
4 +n2. (113)
The solutions of (111) are given by
ˆhk+1/2= Re{CWk} (114)
where C and W are complex-valued constants. By substituting (114) into (111) one finds that
W = Λ±ip
1−Λ2. (115)
W lies on the unit circle when−1≤Λ≤1. The argument ofW, arg(W), is given by
|arg(W)|= cos−1Λ. (116)
When there are K levels in the grid in order to satisfy the boundary conditions, (102) for isopycnal coordinates or
equivalently (106) for height coordinates, it is necessary that
Karg(W) =N π, (117)
whereN is an integer andN−1is equal to the number of times that the mode changes sign within the domain. The solutions are given by
ˆhk+1/2=Csin N πk
K
. (118)
The solution (118) withN =Khasˆhk+1/2= 0 for all integerkwithin the domain so its vertical velocities are zero by (26b). From (105) it also haspˆk = 0for all points in the domain and hence by (22) zero horizontal velocities. When He= 0, (104) does not constrain ρˆand (103) is satisfied provided ρˆk =−ρˆk+1 at all points in the domain. Hence (118) withN =Kcorresponds to the computational mode on the Lorenz grid.
The solutions with N =K−1 are the ones with the smallest equivalent depths that can give rise to Hollingsworth instablities. We now calculate their equiva- lent depths for the realistic case withK >>1. The solution forW for these modes can be expressed as
W = exp
i(K−1)π K
= exp(iπ).exp(−iπ K)
≈ −1 + iπ K.
(119)
Equating (115) and (119) one infers that
p1−Λ2≈ π
K. (120)
AsΛis close to−1, one can take1−Λ2= (1−Λ)(1 + Λ)≈2(1 + Λ)and infer from (120) that
Λ≈ −1 + π2
2K2. (121)
For the isopycnal coordinates (112) then gives
m2= 2(1−Λ)≈4, (122)
whilst (113) gives
n2≈ −4 + 16K2
π2 ≈ 16K2
π2 . (123)
By (108) the equivalent depth,He, is proportional tom−2 so for the height coordinates using the Lorenz grid,Heis a factor of 14π2K−2 smaller than it is for the isopycnal coordinates.
The solutions above are consistent with those derived byTokioka(1978). It is however clearer from the analysis above than that of Tokioka (1978) that the solutions satisfying (120) have the fastest vertical variation and smallest equivalent depths of all the vertical modes that need to be considered on the Charney-Phillips and Lorenz grids.
So all the vertical modes on the Charney-Phillips grid are well-behaved and have equivalent depths of the same order of magnitude as the continuous equations, whilst the modes with the most rapidly variation in the vertical on Lorenz grids have much smaller equivalent depths because of their similarity to the computational mode.
6. Stability matrices and the stability of the modified schemes
6.1. Stability Matrix for the een scheme
The properties of the numerical schemes are best analysed in terms of non-dimensional parameters. So it will be assumed that the perturbations are of a wave-like form
(u′, v′, η′) = (uE, vE, ηE)exp iκx
∆x+iλy
∆y −iωf0t
, (124) where ω is a non-dimensional frequency normalised using f0, and κ and λ are non-dimensional horizontal wavenumbers for thex- andy-directions normalised using the grid spacings∆xand∆y respectively. As is usual in linearised stabilty calculations, physical quantities are given by the real parts of the above expressions and of those
obtained below. Defining
cp= cos(p/2), sp= sin(p/2), p=κ, λ, (125)
for any quantity ψ which varies with x, y and t in the same way as the quantities in (124) thex- andy-averaging operators give
ψx=cκψ, ψy=cλψ, (126)
and the differencing operatorsδxandδygive
δxψ= 2i
∆xsκψ, δyψ= 2i
∆ysλψ. (127) It is convenient also to introduce the coefficients corresponding to the averaging operators (see (128))
µE= 1
3(1 + 2c2κ), νE= 1
3(1 + 2c2λ). (128)
and the associated modified coefficients which are given by
µF =µE, νF =νE for the modified scheme, µF =νF = 1for the original scheme.
(129)
One can define the other non-dimensional quantities in a number of different ways. A convenient approach is to define Froude numbers for the basic flowsu1andv1
Fu= u1
c , Fv= v1
c (130)
and to complete the set of non-dimensional parameters using
Rc= 2c
f0∆y, X= ∆x
∆y. (131)
Rc is twice the ratio of the Rossby radius (c/f0) and the grid spacing ∆y (the factor of 2 has been introduced to simplify expressions later) andX is the ratio of the grid- spacings. In models using latitude and longitude coordinates the latter ratio is small near the pole so the range0< X ≤1 is of interest. The grid-scale Rossby numbersRu andRv
for the flowsu1andv1can be constructed using the above parameters
Ru= 2u1
f0∆x= FuRc
X , Rv= 2v1
f0∆y =FvRc. (132)
We note that the factors of2 in (132) result in values for Ru and Rv that are a factor of 2 larger than the values one would obtain using the classical definition of grid-scale Rossby numbers.
Dividing (65) and (66) byf0, substituting (124) - (132) in them and (67), and writing the result in a matrix form one obtains
i (E11−ω) −cκcλ+ iE12 ig c
Rcsκ X + iE13
cκcλ+ iE21 i (E22−ω) igc(Rcsλ−iE23)
H c
Rcsκ X
H
cRcsλ E33−ω
uE
vE
ηE
= 0.
(133)
where
E11=RusκcκνF+RvsλcλµE, (134) E22=RusκcκνE+RvsλcλµF, (135) E33=RusκcκνF+RvsλcλµF, (136) E12=RvX−1sκcλ(µF−µE), (137) E21=RuXsλcκ(νF−νE), (138) E13=δIFvcκc2λ(µF −µE), (139) E23=δIFuc2κcλ(νF−νE). (140) In (133) the diagonal elementsE11,E22andE33represent advection of u′, v′ and η′ respectively by the basic flow (u1, v1),−cκcλin the same element asiE12represents the Coriolis term−f0v′in (15) andcκcλin the same element as iE21represents the Coriolis termf0u′in (16). Note that the off-diagonal elementsE12,E21,E13andE23are all equal to zero for the modified een scheme.
6.2. Stability of the modified een scheme
The matrix equation (133) can be written in a more symmetric form in two steps. The first step is to multiply the first and second rows of the matrix byiand the last row by−g/c, wherec2=gH. The second step is to multiply the last column of the matrix by H/cand to compensate this by multiplying the last row of the vector byc/H. This gives
ω−E11 −icκcλ−E12 −RcXsκ −iE13
icκcλ−E21 ω−E22 −Rcsλ+ iE23
−RcXsκ −Rcsλ ω−E33
uE
vE cηE H
= 0.
(141)
For the modified een scheme, because E12=E21= E13=E23= 0, the matrix equation (141) has the form
(ωI+H)z= 0, (142)
whereIis the identity matrix andHis an Hermitian matrix, that is a matrix whose transpose is equal to its complex conjugate. All eigenvalues of Hermitian matrices are real- valued and hence the corresponding perturbations are neutrally stable. The eigenvectors of Hermitian matrices are also orthogonal (or can be chosen to be when two or more of the eigenvalues are identical). The gravity wave and Rossby wave solutions of (141) have different phase speeds and hence different eigenvalues so are automatically orthogonal.
In conclusion the linear perturbations of the form (124) can be used to represent any initial conditions and the basic flow is neutrally stable to all linear perturbations.
6.3. Stability matrix for the AL scheme and stability of the modified AL scheme
Let the perturbations again be of the form
(u′, v′, η′) = (uA, vA, ηA)exp iκx
∆x+iλy
∆y−iωf0t
. (143)
