a high resolution balanced scheme for an idealized

Theoretical and Computational Fluid Dynamics manuscript No.(will be inserted by the editor)

A High Resolution Balanced Scheme for an Idealized Tropical

Climate Model

Boualem Khouider1, Andrew J. Majda2

1 Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C., V8W 3P4 Canada

2 Center for Atmosphere/Ocean Sciences, Courant Institute, New York University, 251 Mercer Street, New York, NY

10012, U.S.A.

Received: date / Accepted: date

Abstract. We propose a high resolution balanced numerical scheme for a simplified

tropical climate model with a crude vertical resolution, reduced to the barotropic and

the first baroclinic modes. The two modes exchange energy through highly nonlinear

interaction terms. We consider a periodic channel domain, oriented zonally and centered

around the equator and adopt a fractional splitting strategy, for the governing equations,

into three natural pieces which independently preserve energy. We obtain a scheme which

preserves geostrophic steady states with minimal ad hoc dissipation by using state of art

numerical methods for each piece: The f-wave algorithm for conservation laws with vary-

ing flux functions and source terms of Bale et al. (2002) for the advected baroclinic waves

and the Riemann solver-free non-oscillatory central scheme of Levy and Tadmor (1997)

for the barotropic-dispersive waves. Unlike the traditional use of a fractional splitting

procedure for conservation laws with source terms (here, the Coriolis forces), the class

of balanced schemes to which the f-wave algorithm belongs are able to preserve exactly,

to the machine precision, hydrostatic (geostrophic) steady states. The interaction terms

Correspondence to: B. Khouider (e-mail: [email protected], phone: 1-250-721-7439, fax: 1-250-721-8962)

2 Boualem Khouider et al.

are gathered into a single second order accurate predictor-corrector scheme to minimize

energy leakage. Validation tests utilizing known exact solutions consisting of baroclinic

Kelvin, Yanai, and equatorial Rossby waves and barotropic Rossby wave packets are

given. The overall scheme is then tested on some barotropic-baroclinic solitary waves,

recently discovered by Biello and Majda (2004). The scheme is also tested on a sim-

ple convective parametrization for the propagation of precipitation fronts at large scales

(Frierson et al. 2004) and their interaction with gravity waves, the effect of their ad-

vection by a barotropic trade wind, and their behavior under the beta effect, given the

abundance of diverse equatorial waves with which they could interact.

Key words: high resolution balanced schemes – barotropic-baroclinic nonlinear intera-

tions – precipitation fronts – large scale equatorial waves

PACS: 02.70.Bf, 02.30.Jr,92.60.Bh, 92.60.Dj,92.60.Jq

1 Introduction

There is an increasingly large interest among the atmospheric sciences community regrading the in-

teractions between waves in the tropics and midlatitudes [1–9] as well as the effect of tropical waves

on deep convection. In addition to addressing the issues of global pattern teleconnections between the

equatorial region and the rest of the globe, the study of lateral energy exchange between the tropics

and midlatitudes is also important for understanding the influence of the midlatitudes on tropical wave

dynamics and their interaction with organized deep convective superclusters and many other tropical

phenomena such as monsoons, the Madden-Julian oscillation, and El Nino. Since a significant portion

of the energy in the atmosphere is carried by the barotropic mode which has an important projection

on the midlatitudes, it is meaningful to address this issue through simplified models involving the

interactions between a barotropic flow and the baroclinic-equatorial waves.

The studies of Webster [3–5], Kasahara and Silva Dias [6], Hoskins an Jin [7], etc. revealed the

importance of both a horizontal and a vertical shear for the generation of a midlatitude response

A High Resolution Balanced Scheme for an Idealized Tropical Climate Model 3

from a localized tropical heating source. Majda and Biello [1] have recently derived a simplified model

for the nonlinear interactions between the equatorially trapped and the midlatitude Rossby waves by

a simple Galerkin projection of the beta-plane primitive equations onto the barotropic and the first

vertical baroclinic modes. By using idealized examples, they showed how a barotropic Rossby wave

packet can transfer its energy through a westerly wind burst into equatorially trapped Rossby waves

in the presence of a baroclinic shear and vis versa. Furthermore, theoretical efforts in understanding

convectively coupled tropical waves have focused on models with a crude vertical resolution involving

a dominant baroclinic heating mode coupled to convection [10–15]. Such intermediate models are

important as simplified models for several weeks to seasonal predictions as well as climate studies [16,

17]. They also provide a simplified analytical context to study various strategies for parametrizing

convection [18,14].

Perhaps, the most interesting phenomena in the tropics resides in the remarkable behavior of the

equatorially trapped baroclinic waves [19,20] which involve a mixture of dispersive and non-dispersive

effects as well as their interaction with the barotropic Rossby wave packets in the presence of a vertical

shear, as mentioned above. Also the wave propagation affects a great deal the large scale patterns of

moisture, see, for example, the work of Wheeler and Kiladis in [21] and subsequent papers. Since

the equatorial waves involve balanced as well as unbalanced dynamics, there is a large interest in

developing simplified models which are capable of capturing the appropriate balanced dynamics and

wave interaction across scales. Simplified balanced models used in the literature range from the family

of Weak Temperature Gradient (WTG) models used by Sobel et al. [22] to the systematic multiscale

models of Majda and Klein [23] continuing to the contemporary study by the study by Frierson et al.

[24] the fascinating subject of the interaction of precipitation fronts and equatorial waves at large scales

and the underlying reflection of waves propagating back and forth from a dry to a moist-precipitating

atmosphere, by using a simple convective parametrizations.

The goal of this paper is to design a high resolution balanced numerical model for such intermediate

climate models. We aim to build up a scheme which is very accurate on the scales of interest with

a minimum ad hoc dissipation masking the interaction mechanisms and which is, at the same time,


capable to capture (numerical) geostrophic steady states to machine precision. We use state of the

art numerical methods in an appropriate combination to discretize different pieces of the nonlinear

barotropic-baroclinic model, derived by Majda and Biello [1] following the strategy of Neelin and Zeng

[17] , in a periodic zonal channel centered at the equator, to which we apply a fractional operator

splitting strategy.

The design, the validation, and tests on some idealized situations are found in the remaining parts

of the paper which is organized as follows. In section 2, we (re)derive the simplified model by Galerkin

projecting the nonlinear beta-plane primitive equations onto the barotropic and the first baroclinic

modes. The interacting systems are then augmented and coupled to a vertically averaged moisture

equation. We then show an energy conservation principle for the dry system which is exploited in

section 3 to formulate our basic discretization strategy consisting of fractional operator splitting, for

the governing equations, into three major pieces which conserve their own energies: a system for the

advected equatorial-baroclinic waves, an incompressible system for the dispersive barotropic Rossby

waves, and a third piece gathering the interaction terms into a single system. This is then followed by

a short discussion of the discretization of the interaction system coupling the barotropic and baroclinic

subsystems, which constitute the hard core of the code.

The discretization of the advected baroclinic waves is handled by the f-wave algorithm of Bale et

al. [25] designed for conservation laws with varying flux functions and source terms. This is introduced

and validated in section 4. Particular care is taken for the boundary conditions at the channel walls

to minimize both the wave reflection and the leakage of energy at the wall boundaries. In section

5 we carefully adapt the non-oscillatory central scheme of Levy and Tadmor [26], developed for the

incompressible Euler equations, to discretize the barotropic system with dispersive waves. This is

followed by validation tests using Rossby wave packets while looking at the particular issues associated

with the special case of the channel geometry such as the conditions at the wall boundaries. In section

6 we test the coupled code, in the case of the dry system, on the solitary waves of Biello and Majda [27].

These solutions feature a quite complex flow with both baroclinic and barotropic components resulting

from the resonant interaction of barotropic and equatorially trapped baroclinic Rossby waves.


In section 7, we re-introduce the moisture equation into the system and discuss some associated

exact limiting solutions constructed recently by Frierson et al. [24] in the form of precipitation fronts,

which are moving interfaces separating moist-saturated regions and dry-unsaturated regions, for the

simple case without rotation and without barotropic advection. We then use these solutions to construct

appropriate initial conditions to test the high resolution balanced code developed here for the case with

moisture. Our setup consists of a moving and areally decaying moist region surrounded by two dry

regions; a dying moisture wave (see Figures 10 and 11 below). After the validation of our setup with

a finite (non zero) convective time and the exploration of the interactions of dry gravity waves with

the precipitation fronts, we gradually add the effect of advection by a trade-wind like barotropic wind,

the effect of rotation (beta-effect), and then both rotation and barotropic advection. A concluding

discussion is given in section 8 while some technical details concerning the design of a Poisson solver

associated with the incompressible central scheme in a channel as well as the algorithm leading to the

dying moisture wave are described in appendices I and II, respectively, at the end.

2 The simplified climate model

2.1 The barotropic-first baroclinic model with moisture

We consider the primitive equations on a beta-plane with rigid lid boundary conditions.

(A)DVH

Dt+ βyV⊥

H = −∇HP + Sv

(B) divHVH +Wz = 0

(C)DΘ

Dt+N2θ0g

W = SΘ (2.1)

(D)∂P

∂z= g

Θ

θ0

(E) W |z=0,HT = 0.

Here VH = (U, V ) is the horizontal velocity, W is the vertical velocity, P is the pressure, and Θ is

the potential temperature. Capital letters are used in order to distinguish quantities represented by

the primitive equations from their Galerkin projection components which are introduced below. Here


N = 0.01 s−1 is the Brunt-Vaisala buoyancy frequency, g = 9.8 m s−2 is the gravitational acceleration,

and θ0 = 300 K is a reference (constant background) potential temperature. β = 2.2804 × 10−11

s−1m−1 is the gradient of the Coriolis force at the equator. Also

D

Dt≡ ∂

∂t+ U

∂

∂x+ V

∂

∂y+W

∂

∂z

is the material derivative while divH ,∇H are the horizontal divergence and horizontal gradient, respec-

tively while V⊥H = (−V, U) and Sv, Sθ represent sources and sinks of momentum and heat respectively,

e.g., Rayleigh friction and convective heating and/or radiative cooling. Typically, the detailed moist

dynamics are obtained by coupling the equations in (2.1) to the bulk cloud microphysics equations

invoking the budget equations of the moisture variables such as cloud water, water vapor, and rain (see

for eg. [24] for details). Here, for simplicity, we skip this and present rather directly the equation for the

vertically averaged total water content, q, coupled to the barotropic-first baroclinic equations, which

are presented next. The barotropic-first baroclinic equations were derived by Majda and Biello [1] by

a simple vertical Galerkin projection of the nonlinear primitive equations in (2.1) onto the barotropic

and the first baroclinic modes (see [20] for details) following the strategy of Neelin and Zeng [17]. Re-

call that the barotropic mode is obtained by a simple vertical averaging over the tropospheric height,

HT ≈ 16 km, of the equations in (2.1) while the projection onto the first baroclinic mode is achieved

by making the functional inner product,

〈f, g〉 =1

HT

∫ HT

0

f(z)g(z)dz,

with g = cos

(

zπ

HT

)

for the case of the horizontal velocity, VH , and the pressure, P , and g = sin

(

zπ

HT

)

for the potential temperature, Θ, and the vertical velocity, W , i.e, the following crude vertical approx-

imation is made

VH

P

(x, y, z, t) ≈

v

p

(x, y, t) +

v

p

(x, y, t) cos(πz

HT)

W

Θ

(x, y, z, t) ≈

w

θ

(x, y, t) sin(πz

HT). (2.2)

Notice that because of the rigid-lid condition in (2.1) (E) and the hydrostatic balance in (2.1) (D),

the barotropic components of both the vertical velocity and the potential temperature vanish. The


hydrostatic equation links the baroclinic components of pressure an temperature,

p = −gHT θ

πθ0,

while the vertical velocity is recovered through the continuity equation,

w = −HT

πdivv.

The various dynamical variables, v, p,v, θ, are converted to non dimensional quantities by using

the following reference scales, relevant to the tropical wave dynamics [20,19,15, etc]: the gravity wave

speed c = NHT /π ≈ 50 m/s is used as the velocity scale, the equatorial Rossby deformation radius L =

(

cβ−1)1/2 ≈ 1500 km as the length scale, T = L/c ≈ 8 hours as the time scale, and α = HT N2θ0

πg ≈ 15

K is our temperature scale. Therefore, the barotropic-baroclinic nonlinear interacting system resulting

from the Galerkin projection strategy above, in nondimensional units, simplifies to

∂v

∂t+ v · ∇v + yv⊥ + ∇p = −1

2(v · ∇v + vdivv)

= −1

2div (v ⊗ v)

divv = 0 (2.3)

and

∂v

∂t+ v · ∇v −∇θ + yv⊥ = −v · ∇v

∂θ

∂t+ v · ∇θ − divv = S(x, y, t) (2.4)

The momentum friction/forcing terms are ignored for simplicity and S(x, y, t) denotes the projection

of SΘ in (2.1) onto the first heating mode.

In order to get a closure for the source term, S(x, y, t), in (2.4), we introduce next the equation

governing the vertically integrated moisture content, q, with a prescribed background moisture gradi-

ent.

∂q

∂t+ v · ∇q + Qdivv = −P +E (2.5)

where P is the precipitation rate, E is the surface evaporative flux, and

Q = QLc2HT

π2, (2.6)


where Lc ≈ 188 is a dimension-less rescaling constant involving the latent heat of vaporization which

permits one to convert moisture quantities into temperature units and Q is the actual background

moisture gradient, imposed over a moisture scale height which is taking here to be equal to the total

tropospheric height, HT . For a detailed discussion on (2.5), we refer the interested reader to [24]. A

rigorous derivation involving an arbitrary moisture scale height will be published elsewhere by the

authors.

The system formed by (2.3), (2.4) is coupled to the equation in (2.5) through the source term

S and the precipitation rate, P , which are parametrized together as functions of both the potential

temperature and the moisture content variables θ and q. Typically, we write

S = C −QR (2.7)

where C is the convective heating which is proportional to the precipitation rate P and QR is a

Newtonian cooling term. Below and in the following four sections we drop the moisture equation

and with it the convective forcing terms and concentrate ourselves on the design of a high resolution

balanced numerical scheme for the dry system formed by (2.3) and (2.4). We shall return to the

moisture equation and to the convective parametrization in section 7.

2.2 Conservation of total energy for the dry system in a channel

We consider the dry system formed by (2.3), (2.4) with S ≡ 0 in zonally oriented periodic channel.

The north-south walls are located at a distance Y = 5000 km away from the equator and the zonal

period is equal to the perimeter of the earth at the equator, i.e, X = 40 000 km. We assume a no-flow

boundary condition at the channel walls for both the barotropic and the baroclinic flows, to contain

the fluid inside the channel.

v(x,±Y, t) = 0, v(x,±Y, t) = 0. (2.8)

Let

EC(t) =1

4

∫ Y

−Y

∫ X

0

|v|2 + θ2 dxdy (2.9)


be the total (kinetic + potential) energy associated with the baroclinic flow and

ET (t) ≡ 1

2

∫ Y

−Y

∫ X

0

|v|2 dxdy (2.10)

be the kinetic energy associated with the barotropic flow. Note that the peculiar factor of 1/4 in the

total energy in (2.9) arises from the fact that the vertical projection omits√

2 and is therefore is not

a unit eigenvector. Then, under the boundary conditions in (2.8) , the total (dry) energy defined by

Ed(t) ≡ EC(t) + ET (t) (2.11)

is conserved. In fact, we have

d

dtEC(t) = −1

2

∫ X

0

∫ Y

−Y

v ⊗ v · ∇v dxdy.

and

d

dtET (t) = −1

2

∫ X

0

∫ Y

−Y

v · div (v ⊗ v) dxdy.

This leads to

d

dtEd(t) = −1

2

∫ X

0

∫ Y

−Y

div [(v ⊗ v)v] dxdy = 0 by (2.8)

Notice that if the interaction terms on the right hand-sides of (2.3) and (2.4) are zero, then the

last two time derivatives above vanish, i.e, both the baroclinic and the barotropic systems in (2.3)

and (2.4), respectively, conserve their own energy when their mutual interaction is ignored. Therefore,

the energy of interaction is also conserved. This is what is actually exploited below to design a high

resolution balanced scheme with minimum ad hoc dissipation.

3 The basic discretization strategy

3.1 Time splitting algorithm in continuous form

Now we take advantage of the way the coupled system in (2.3) and (2.4) conserves the different

forms of energy, as discussed above, to design a time splitting (fractional step) algorithm where the

baroclinic-barotropic interaction system is broken onto three pieces which individually conserve energy.

The resulting subsystems are then discretized and coded into three different routines which are called

separately in the code via the Strang methodology. The resulting systems are given next, in continuous

form. They consist of


– a linear conservation system for the advected baroclinic waves

∂v

∂t+ div (v ⊗ v) −∇θ + yv⊥ = 0

∂θ

∂t+ ∇ · (vθ) − divv = 0, (3.1)

– a nonlinear incompressible system for the barotropic waves

∂v

∂t+ v · ∇v + yv⊥ = −∇p

∇ · v = 0, (3.2)

– and a third system gathering the coupled nonlinear interaction terms into a single coupled system.

Below, the latter is written in terms of the potential vorticity.

3.2 Remarks on the partial discretizations and on the numerical boundary conditions

Notice that because the barotropic-advecting flow is incompressible, the system in (3.1) is alternatively

written in conservative form. This will also be exploited, below, for the barotropic piece in (3.2). The

discrete system resulting from (3.1) is closed with the aide of the periodic boundary conditions in x

and the homogeneous Dirichlet boundary conditions at y = ±Y which are compatible with the no-flow

condition in (2.8) as well as no-reflecting and extrapolation with the exact solution, for validation

purposes, as described below in section 4.

In order to conserve geostrophy (numerically), for the barotropic system in (3.2), we pass to the

potential vorticity-stream function formulation. We introduce the potential vorticity

ξ = vx − uy + y

and the stream function

u = −ψy; v = ψx.

The system in (3.2) leads to

∂ξ

∂t+ u

∂ξ

∂x+ v

∂ξ

∂y= 0

∆ψ = ξ − y (3.3)

u = −ψy; v = ψx.


Since, according to the first equation in (3.3) and the divergence constraint, the potential vorticity, ξ, is

conserved, numerical geostrophic balance for the barotropic flow is automatically guaranteed provided

we use a Godunov-type scheme to discretize the conservation law.

The continuous system in (3.3) is seemingly closed by the periodic boundary conditions in x and

the no-flow (v = 0) condition at the channel walls. However, for the discrete version, some numerical

values are needed at the ghost cells outside the walls, for the potential vorticity, ξ, and they have

to be ’chosen’ carefully. Particularly, we demonstrate in section 5, below, that homogeneous Dirichlet

boundary conditions do a better job than the popular polynomial extrapolation ones which tend to

create a numerical boundary layer near the walls where vorticity is generated and is amplified in time.

In terms of the vorticity-stream function formulation, we have for the interaction system

∂ξ

∂t= −1

2

((

∂2

∂x2− ∂2

∂y2

)

(uv) +∂2

∂x∂y

(

v2 − u2)

)

(3.4)

∂v

∂t= −v · ∇v

∆ψ = ξ − y

u = −ψy; v = ψx, v = (u, v), v = (u, v)

For compatibility, when discretized, the system in (3.4) is closed by the same boundary conditions as

inherited from (3.1) and (3.3).

Recall that at each fractional step, only one among the three sub-systems in (3.1), (3.3), and (3.4) is

integrated independently while the others are frozen. Especially, the integration of the system in (3.1),

from time t to time t+∆t, where ∆t > 0 is the time step, common to the three systems, assumes that

the advecting-barotropic flow (u, v) is constant on the interval [t, t+∆t]. This is consistent, by many

means, with the use of the f-wave method for conservation laws with space dependent flux functions

but autonomous in time, in section 4, for the baroclinic system. This method is introduced by LeVeque

and his collaborators [25] to handle flux-varying conservation laws with source terms. It leads to a high

resolution geostrophically balanced second order numerical scheme for the advected baroclinic waves

system in (3.1). For the potential vorticity-stream function system in (3.3) we use the non-oscillatory


central scheme, introduced by Levy and Tadmor in [26] for the 2d incompressible Euler equations (see

[28] for the version which deals directly with the momentum formulation).

Before moving forward to discussing the details of the numerical schemes used for solving the

subsystems in (3.1) and (3.3), which constitute the hard-core of the present code, we will first design

a second order scheme for solving the energy conserving interaction system in (3.4). For this latter, we

use central finite differences in space coupled with a predictor corrector method in time.

3.3 A second order discretization for the interaction system

The spatial derivatives on the right hand side of the evolution equations in (3.4) are discretized by

centered differences, using a five point stencil so that

∂2f

∂x2≈ fi+1,j − 2fi,j + fi−1,j

∆x2,∂2f

∂y2≈ fi,j+1 − 2fi,j + fi,j−1

∆y2,

and∂2f

∂x∂y≈ fi+1,j+1 − fi−1,j+1 − fi+1,j−1 + fi−1,j−1

4∆x∆y. (3.5)

The discrete incompressible barotropic flow is tied to the evolving potential vorticity, ξi,j , through

the discrete stream function, ψi,j , by a five-point Poisson solver, as discussed in Appendix I. The

discrete barotropic velocity field is obtained by 2nd order centered differences so that it remains

incompressible as in section 5 below. The time integration is performed a 2nd order Runge-Kutta type

predictor-corrector method, applied to the semi-discrete version of (3.4), obtained by substituting the

finite differences in (3.5) into the spatial derivatives.

3.4 Alternate discretization

Before we end this section, it is worthwhile mentioning that other-alternate splitting strategies are

possible. A somewhat natural splitting strategy, which will also yield energy conserving subsystems,

would be obtained by moving the advection terms in (3.1) into the system in (3.3) so that the new

system, which is accordingly augmented with the advection equations for v and θ, would be overall

solved by the central scheme of Levy and Tadmor (in section 5); leaving a simple linear shallow

water system which can be solved by any well balanced solver, eg. [29], [30], [31]. Nevertheless, the


splitting strategy keeping the advection terms in (3.1) is justified in part by the facts that 1) the

f-wave algorithm in [25] is especially designed to handle conservation laws with varying fluxes, hence,

especially, advection systems, 2) the flows of interest are ’subsonic’ (the velocity maximum is below

the gravity wave speed) so that the system in (3.1) remains strictly hyperbolic, and more importantly

3) we want to track the geostrophic balance with respect to a background advecting barotropic flow

as well.

4 A balanced high resolution method for equatorial baroclinic waves

As mentioned above we use the f-wave algorithm of Bale et al. [25] to solve the advected shallow water

system in (3.1). This results in a robust high resolution 2nd order scheme with the property that

initial geostrophic steady states will remain so during the discrete integration; even in the presence of

a barotropic wind.

The f-wave algorithm came as a generalization and an improvement, toward conservation laws with

varying flux functions, of the quasi-steady wave-propagation method proposed by LeVeque in [29] to

compute (nonlinear) shallow water flows which are near hydrostatic balance with respect to topographic

source terms. A more sophisticated approach, which as well conserves dry beds, is introduced and used

in [30], [31]. The scheme in [30] makes the systematic use of left and right states to solve the Riemann

problem directly while LeVeque’s wave propagation method is based on the flux differences, i.e., it

assumes a Roe-like numerical scheme. Furthermore, the high resolution version of LeVeque scheme is

obtained through the flux-limiting strategy while the former uses the slope-limiting technique. However,

because the f-wave algorithm is originally designed and easily implemented for one-space-dimension

systems, (3.1) is further split onto two unidimensional systems for each direction x and y, respectively.

4.1 Dimensional operator splitting

It is common to use the dimensional operator splitting strategy for the multi-dimensional shallow

water equations such as (3.1) [29,31, etc], permitting to advance in time each direction separately.

This is done in such fashion that the Coriolis terms are distributed onto the resulting 1d systems so


that both the meridional and zonal velocities, v, u, are decoupled from the x- and y-systems into two

simple linear-advection equations in the x- and y-directions, respectively. Namely, (3.1) is decomposed

onto the following two 1d systems:

∂u

∂t+∂(uu)

∂x− ∂θ

∂x− yv = 0

∂θ

∂t+∂(uθ)

∂x− ∂u

∂x= 0 (4.1)

∂v

∂t+∂(uv)

∂x= 0

and

∂v

∂t+∂(vv)

∂y− ∂θ

∂y+ yu = 0

∂θ

∂t+∂(vθ)

∂y− ∂v

∂y= 0 (4.2)

∂u

∂t+∂(vu)

∂y= 0.

Notice that in order to guarantee second order accuracy a Strang-splitting procedure is needed when

alternating the different pieces of the code. Furthermore, it is demonstrated, for the case of the equa-

torial waves here (results not shown), that the Strang-splitting procedure, as opposed to the classical

fractional step method which is only first order is required even for the directional splitting in (4.1) and

(4.2), in order to recover the second order accuracy. Particularly, the L1 error is improved by one order

of magnitude on the coarse mesh of 128 X 75 grid points. This finding is in itself interesting because it

is often argued in the literature that the Strang’s strategy is not necessary for the dimensional splitting

on the grounds that this latter induces only small errors which are negligible compared to the errors

induced by the partial discretizations of the 1d systems.

4.2 The f-wave algorithm with source term

The f-wave algorithm is introduced in [25] as an improvement and a generalization of the wave prop-

agation algorithm in [29] to handle conservation laws with spatially varying flux functions and source

terms (balance laws). The f-wave algorithm as opposed to the original wave propagation algorithm

decomposes the flux increments, F (UI+1) − F (Ui), directly into simple waves instead of passing by


the decomposition of the increments, Ui+1 − Ui, as the original algorithm does. Because it is actually

the flux increments which are needed and which are used within the code for the updating of the

state variables and not the wave increments. The new algorithm avoids the unnecessary calculations

permitting to pass from the individual waves to an approximation of the flux differences which makes

it more robust and more efficient. It has the same implementation pattern as the original wave prop-

agation algorithm including its second order extension. Perhaps the most important advantage of the

f-wave algorithm resides in its ability to handle balanced laws near steady states with high accuracy.

It incorporates the source terms into the flux increment before the wave decomposition is performed.

Therefore, it is only the deviation of the (computed) solution from steady state which is decomposed

into waves so that steady states are automatically preserved since, in this case, the strength of the

waves resulting from a (numerically) balanced state would be zero. For a detailed discussion, the inter-

ested reader is referred to [29], [25], and references therein. Nevertheless, we give next a brief summary

for the sake of completeness.

To illustrate, we consider the following generic conservation law system with source term (balance

law).

∂U

∂t+∂F (x, U)

∂x+B(U) = 0. (4.3)

Here F (x, U) is a vector valued spatially varying flux function pointing the conservative part of either

one of the systems in (4.1) and (4.2) while the term B(U) points toward the Coriolis force terms −yv

and yu, respectively. Since the f-wave algorithm is based on the wave propagation method, it is easier

to understand the former by first giving a short outline of the latter.

4.3 Wave propagation algorithm for conservation laws

The wave-propagation algorithm for a conservation law,

Ut + F (U) = 0,

consists on rewriting the left and right fluxes, F (U ∗

i±1/2), of the associated finite-volume discretization,

Un+1

i = Uni − ∆t

∆x

[

F (U∗

i+1/2) − F (U∗

i−1/2)]

, (4.4)


with U∗

i−1/2is the solution to the Riemann problem at the interface i − 1/2 with the left and right

states Ui−1 and Ui, respectively, as a linear combinations of right- and left-going simple waves. The

resulting algorithm is summarized below through the following abstract formulation:

Un+1

i = Uni − ∆t

∆x

[

A+(∆Ui−1/2) + A−(∆Ui+1/2)]

(4.5)

where A±(∆Ui−1/2) represent respectively the left going (−) and a right going (+) fluxes across the

edge i− 1/2 with the flux increment

F (Ui) − F (Ui−1) = A+(∆Ui−1/2) + A−(∆Ui−1/2)

and

∆Ui−1/2 = Ui − Ui−1

is the wave increment or the jump across i− 1/2. Notice that A+(∆Ui−1/2) changes the values of the

vector Ui while A−(∆Ui−1/2) changes the values of Ui−1. For a hyperbolic system the jump Ui −Ui−1

is decomposed onto M waves (typically M = n, the dimension of the system), Wpi−1/2

, p = 1, 2, · · ·M

propagating with the speeds λpi−1/2

, p = 1, 2, · · · ,M :

Ui − Ui−1 =M∑

p=1

Wpi−1/2

,

which yields

F (Ui) − F (Ui−1) =M∑

p=1

λpi−1/2

Wpi−1/2

.

Here λpi−1/2

are the eigenvalues of (some approximation of) the Jacobian matrix ∂F (U)/∂U (Roe

Matrix) and the waves Wpi−1/2

are the projection onto the associated eigenvectors. The flux-difference

scheme in (4.5) is then closed by setting

A−(∆Ui−1/2) ≡M∑

p=1

(λpi−1/2

)−Wpi−1/2

A+(∆Ui−1/2) ≡M∑

p=1

(λpi−1/2

)+Wpi−1/2

(4.6)

where λ+ = max(λ, 0) and λ− = min(λ, 0).

The second order extension in the context of the wave algorithm is obtained by adding on the right

hand side of the updating formula in (4.5) the appropriate wave correction terms, yielding [29]

Un+1

i = Uni − ∆t

∆x

[

A+(∆Ui−1/2) + A−(∆Ui+1/2)]

− ∆t

∆x(Fi+1/2 − Fi−1/2), (4.7)


where the second order correction Fi−1/2 is given by

Fi−1/2 =1

2

M∑

p=1

|λpi−1/2

|(

1 − ∆t

∆x

∣

∣

∣λpi−1/2

∣

∣

∣

)

Wpi−1/2

with

Wpi−1/2

= limiter(Wpi−1/2

,WpI−1/2

), I =

i− 1 if λpi−1/2

> 0

i+ 1 if λpi−1/2

< 0.

Here limiter is the flux limiting function leading to a high resolution scheme, which is specified in the

code. We actually use the monotonized central (MC) limiter (see for e.g. [32]).

4.4 F-wave algorithm for conservation laws

The f-wave algorithm uses the same methodology as the wave-propagation algorithm of decomposing

into individual waves the left and right going fluxes at the cell edges i+ 1/2. The main and somewhat

the only technical difference between the two schemes resides in the fact that the new algorithm uses

instead a decomposition of the flux increments directly. Namely, we let

F (Ui+1) − F (Ui) =

M∑

p=1

βpRp =

M∑

p=1

Zpi+1/2

where Rp’s are the eigenvalues of the approximate Jacobian matrix, mentioned above, at the cell

edges i + 1/2. Nevertheless, the new algorithm yields a conservative scheme even if the associated

Roe scheme is not conservative and it is especially designed to perform on conservation laws with

varying flux functions, as it is the case for the advected baroclinic systems in (4.1) and (4.2). More

importantly, for the case of balanced laws, the f-wave algorithm leads easily into a balanced scheme by

directly incorporating the source term into the flux differences at the cell interfaces, before the wave

decomposition is performed, as summarized below.

To some extent, the f-wave algorithm can be viewed as an new way of rewriting the original wave

propagation algorithm. This is true at least for the linear case for which we have

F (Ui) − F (Ui−1) =

M∑

p=1

Zpi−1/2

=

M∑

p=1

αpλi−1/2Rpi−1/2

=

M∑

p=1

λpi−1/2

Wpi−1/2

with αp =

βp

λp, λp 6= 0

0, λp = 0

p = 1, 2, · · ·M.


Therefore its implementation follows directly from the wave propagation algorithm and particularly

the second order flux corrections are given by

Fi−1/2 =1

2

M∑

p=1

(

1 − ∆t

∆x|λp

i−1/2|)

sign(λpi−1/2

)Zpi−1/2

.

4.5 F-wave algorithm for balance laws

In the presence of source terms like in (4.3), the f-wave algorithm leads easily to a balanced scheme

where analytical steady states are captured by the numerical scheme to the machine precision. The

source term is first distributed onto delta functions defined on the cell edges so that it is rather the

flux difference plus the source term, as a whole, which is decomposed into waves

F (Ui+1) − F (Ui) +∆xB(Ui+1/2) =

M∑

p=1

Zpi+1/2

where B(Ui+1/2) is some approximation of B(U) at the cell edges i+ 1/2 (e.g. B(Ui+1/2) = (B(Ui) +

B(Ui+1))/2). In this fashion, a (discrete) steady state (F (Ui+1)−F (Ui) +∆xB(Ui+1/2)) will result in

a decomposition into waves with zero strengths.

In order to get a second order scheme for the balanced scheme, the following correction term needs

to be added into the updating formula in (4.7).

(∆t)2

2∆x

∂B(Ui)

∂U

M∑

p=1

Zpi+1/2

+ Zpi−1/2

2.

4.6 Numerical boundary conditions at the walls

While the periodic boundary conditions for the x-direction system in (4.1) are easily implemented, the

handling of the boundary conditions at the artificial walls during the implementation of the y-direction

system in (4.2) is a serious numerical issue. Below we propose and, later on, compare three types of

boundary conditions for (4.2).

The first type of boundary conditions we consider here is based on a wave-non-reflection principle

which consists of not allowing outgoing waves to be reflected into the domain. This type of boundary

conditions are particularly suitable when artificial walls are imposed within the body of the fluid to

delimit the computational domain which is otherwise infinite. In order for a one way-wave equation


to be well-posed, no boundary condition should be imposed on the artificial wall where the wave is

going outward but a boundary condition is required at the wall where the information is inward [33].

It is this idea which is exploited and expanded to the shallow water equations by the aide of the

Riemann invariants [33]. Since by (2.8) the barotropic meridional wind v is set to zero at the walls,

(4.2) simplifies to

∂v

∂t− ∂θ

∂y+ yu = 0

∂θ

∂t− ∂v

∂y= 0 (4.8)

∂u

∂t= 0.

at y = ±Y . The last equation suggests that, except for variations due to the x-system in (4.1), u is

constant in time at the channel walls. Hence any type of boundary conditions can be imposed on u

when solving (4.2). Next we introduce the Riemann invariants for the system formed by v and θ,

r+ = v − θ; r− = v + θ

which satisfy

∂r±

∂t± ∂r±

∂y+ yu = 0. (4.9)

Given r± at time t = 0, the exact solutions for (4.9) at times t > 0 is given by [34]

r±(y, t) = r±(y ∓ t, 0) −∫ t

0

(y ∓ (s− t))u(y ∓ (s− t), s)ds;

If we use a zero order extrapolation for u to the ghost cells outside the walls, then from the exact

formula for r+ above we get, for example at the south wall,

r+(y, t) = r+(y − t, 0) − u(−Y )

∫ t

0

(y − (s− t))ds = r+(y − t, 0) − u(−Y )(yt+1

2t2)

By imposing the radiation condition r+(., 0) ≡ 0 outside the domain so that no energy is injected into

the domain we get

r+(y, t) = −u(−Y )(yt− t2)

and since r− is propagating inward, we make the extrapolation r−(y, 0) ≡ r−(−Y, 0), for y < −Y , we

get

r−(y, t) = r−(−Y, 0) − u(−Y )(yt− 1

2t2).


By using similar arguments at the north wall, we get the following

BC#1: Non-reflecting boundary conditions

u(y, t) = u(−Y, 0), r+(y, t) = −y∆tu(−Y, 0),

and r−(y, t) = r−(−Y, 0) − y∆tu(−Y, 0) if y < −Y

u(y, t) = u(Y, 0), r+(y, t) = r+(Y, 0) − y∆tu(Y, 0),

and r−(y, t) = −y∆tu(Y, 0) if y > Y (4.10)

within the time step ∆t where we dropped the terms involving the second powers of t.

The second type of boundary conditions we are interested in is the homogeneous (fixed) Dirichlet

boundary conditions. This type of boundary conditions is motivated by the solid wall conditions

imposed in (2.8) to ensure energy conservation for the channel geometry.

BC#2: Homogeneous Dirichlet BC for all the variables, u, θ, v:

u(y) = 0, v(y) = 0, θ(y) = 0, if y < −Y or y > Y. (4.11)

Finally, the third type of boundary condition of interest here consist in fixing the values of u, θ, v

to the exact analytical solution. This is of course impractical because we are not supposed to know

the solution of the problem before hand but it turned out to be very useful when it comes to making

validation tests.

BC#3: Exact solution for validation

The numerical solution at the ghost cells outside the walls is fixed to its exact

analytical value there, i.e, Dirichlet boundary condition using the exact solution.

4.7 Validation tests

Now we propose to validate the balanced algorithm proposed above for the linear beta-plane equatorial

shallow equations by using the large scale equatorial waves, well known among the atmospheric science

community [19,20]. These are exact solutions for the baroclinic system in (3.1) in the absence of the


barotropic advection (v ≡ 0). Especially, we demonstrate that the f-wave scheme is capable to capture

accurately the geostrophic balanced as well as the unbalanced waves. In particular, we consider the

following large scale equatorial waves [19,20]. The Kelvin wave,

u(x, y, t) = cos(2πx− t

X)e−y2/2, v(x, y, t) = 0, θ(x, y) = −u(x, y, t), (4.12)

where X is the perimeter of the equator expressed in non-dimensional units, which is perfectly

geostrophically balanced in the meridional direction since the meridional flow, v, is zero. The Yanai

wave, which is given by

v(x, y, t) = cos(kx− ωt) exp(−y2/2);

u(x, y, t) = −ωy exp(−y2/2) sin(kx− ωt); (4.13)

θ(x, y, t) = − 1¯αu(x, y, t))

where ω = ( 2πX +

√

4π2

X2 + 4)/2 is the phase of the wave and k = 2π/X is the wavenumber, has a strong

cross equatorial flow thus it provides a (strongly) unbalanced test case. Finally, the anti-symmetric

equatorial Rossby wave of index 2 (MI= 2),

v(x, y, t) = (2y2 − 1)e−y2/2 cos(kx− ωt)

r+(x, y, t) =4y3 − 6y

k − ωe−y2/2 sin(kx− ωt) (4.14)

r−(x, y, t) = − 4y

k + ωe−y2/2 sin(kx− ωt)

with ω is the smallest in magnitude root of the dispersion relation

ω2 − k2 − k

ω= 5

and r+ = u−θ, r− = u+θ are the Riemann invariants; this example is chosen because it constitutes

a weakly balanced solution according to the long wave approximation theory [19,20]. Furthermore,

notice that the Kelvin wave is non-dispersive while the Yanai and MI=2 Rossby waves are dispersive

and the Kelvin wave and Yanai wave, as regards the meridional velocity v, are symmetric with respect

to the equator while the MI=2 Rossby wave is anti-symmetric. Thus, these three exact solutions form

a set of tests which is representative of the equatorial baroclinic (linear) dynamics in the sense that if

the numerical scheme behaves well for these three cases it will capture with the comparable accurately

the balanced/unbalanced dispersive/non-dispersive large scale dynamics in the tropics.


In Table 1, we compare the exact solutions, ue, ve, θe, as given in (4.12), (4.13), or (4.14), respectively,

with their numerical analogues, un, vn, θn, computed by the f-wave algorithm using the exact boundary

condition in BC#3.

We report the L1-norm (∫ ∫

|un−ue|+ |vn −ve|+ |θn −θe|dxdy), after an integration time of about

2 days for the Kelvin and Yanai waves and about 47 days for the MI=2 Rossby wave. Three different

grids, 128 × 75, 256 × 150, and 512 × 300 with the CFL number .9 are used. Since these solutions do

not develop shocks or discontinuities, the second order accuracy is established with and without (not

shown) the MC limiter.

[Table 1 about here.]

Figure 1 displays the time history of the baroclinic energy for the three runs on the last row, for the

MI=2 Rossby wave. (The energy plots for the other waves resemble to Figure 1, therefore there are

not shown here). Figure 1, particularly, demonstrates that except for some small numerical dissipation

which diminishes with the grid refinement (on the order two at least), the scheme conserves energy.

Before closing this subsection, it is worthwhile mentioning that error spatial-distribution plots (not

shown) showed that the local errors dominate within the interior of the domain at the locations where

the wave topological changes are more significant.

[Fig. 1 about here.]

4.8 Role of boundary conditions and distance from walls

As noted above the boundary condition BC#3 assuming the knowledge of the solution at the bound-

aries can be used only for validation purposes. Here, we investigate the effect of more practical boundary

conditions such as the ones given in BC#1 and BC#2 for the computations of the equatorial waves in

a channel. Also we check the sensitivity of the results to the distance between the north-south walls by

increasing Y from the more physical value of Y = 5, 000 km to a more conservative value Y = 8, 000


km. For brevity we consider only the case of the first symmetric (MI=1) Rossby wave,

v(x, y, t) = ye−y2/2 cos(kx− ωt)

q(x, y, t) =2y2 − 1

k − ωe−y2/2 sin(kx− ωt) (4.15)

r(x, y, t) = − 1

k + ωe−y2/2 sin(kx− ωt)

with ω is the smallest in magnitude root of the dispersion relation

ω2 − k2 − k

ω= 3.

Notice that this constitutes an intermediate new test which is harder compared to the Kelvin and

Yanai waves because it is less trapped near the equator but perhaps easier than the MI=2 case.

In table 2, we report the L1-norm errors occurred between the exact solution and the numerical

solution using the boundary conditions BC#1, BC#2 and BC#3 as well as the results obtained with

the boundary conditions BC#2 when the wall distance is increased to Y = 8000 km, for the integration

time of about 16 days equivalent to one period. Again the second order convergence is observed with

the use of the exact boundary condition BC#3.


However, the rows BC#1 and BC#2 with Y = 5000 km, in Table 2, demonstrate particularly

that the solutions using the non-exact boundary conditions do not converge to the expected equatorial

(M=1) Rossby wave as the error does not decrease with the grid refinement. Nevertheless, the L1-error

remains small and as shown on Figure 2 (A) the tiny errors are confined to a small strip (numerical

boundary layer) near the walls. More importantly, as shown on Figure 2 (B), the energy remains

essentially conserved except for a small transient period where a small deviation from the initial

condition due to the use of non-exact boundary conditions occurs. Moreover, the use of a larger

wall distance (Y = 8000 km) ’circumvents’ this boundary condition artifact. In fact, the last row

in table 2 ’demonstrates’ a second order convergence and the energy plots on Figure 2 (D) show

energy conservation from the beginning. (Except for the expected numerical dissipation, etc. which is

nevertheless small and converges to zero with the order two, with the grid refinement.)



The error distribution mesh plot on Figure 2 (C) shows no numerical boundary layer unlike the

panel (A) above it. This clearly demonstrates the role played by the distance from the walls on the

approximation of the equatorial waves when using ’artificial’ boundary conditions such as in BC#1

and BC#2. Because the equatorial waves are trapped at the equator, the further the walls are from

the equator the less the discrepancies between the boundary conditions which are used (e.g. Dirichlet)

and the exact solution are important, compared to the second-order error of discretization occurring

inside the domain.

5 High resolution method for barotropic waves

For the barotropic equations in (3.2), we propose to adapt the central scheme introduced and used in

[26] by Levy and Tadmor for the 2d incompressible Euler equations. The novelty here is that, unlike

the Euler equations, (3.2) supports dispersive (Rossby) waves. By analogy to what Levy and Tadmor

did, we deal directly with the potential vorticity-stream function formulation in (3.3). As noted in

[26], the design of the central-incompressible scheme takes a double advantage in the facts that the

dynamical equation for the (potential) vorticity can be alternatively written in an advective form as

in (3.3) or in the conservative from,

∂ξ

∂t+

∂

∂x(uξ) +

∂

∂y(vξ) = 0, (5.1)

thanks to the incompressibility constraint. The conservative form justifies the use of a Godunov type-

finite volume method while the advective form guarantees a finite-speed of propagation. The details

of the method are given in [26], we next give a brief summary for the sake of completeness.

5.1 The 2d central scheme

To illustrate, we consider the generic scalar conservation law in two space dimensions,

ut + f(u)x + g(u)y = 0, (5.2)


where, u is the unknown scalar solution (to not confuse with the baroclinic zonal velocity component)

and f and g are the fluxes in x and y directions, respectively.

Let xj , yk, for the integers j, k, be a uniform discretization of the x-y plane with the mesh sizes ∆x

and ∆y respectively. We also assume a time discretization such that tn+1 = tn +∆t, n ≥ 0 with ∆t ≥ 0

is the time step to be specified latter. At each discrete time t = tn, n ≥ 0, we define the piecewise

polynomial function

w(x, y, tn) =∑

j,k

pnj,kχj,k(x, y), (5.3)

as an approximation-reconstruction of the solution u in (5.2). χj,k is the indicator function of the

rectangular cell Cj,k of length ∆x×∆y centered at xj , yk. By analogy, the staggered cell, Cj+1/2,k+1/2,

is the rectangle of length ∆x ×∆y and center xj+1/2, yk+1/2. The finite volumes method applied to

(5.2) on the staggered cells leads to

un+1

j+1/2,k+1/2≈ wn+1

j+1/2,k+1/2=

1

∆x∆y

∫ ∫

Cj+1/2,k+1/2

w(x, y, tn) dxdy (5.4)

+1

∆x∆y

∫ yk+1

yk

∫ tn+1

tn

f(w(xj+1, y, τ)) − f(w(xj , y, τ)) dydτ

+1

∆x∆y

∫ xj+1

xj

∫ tn+1

tn

g(w(x, yk+1, τ)) − g(w(x, yk , τ)) dxdτ

By using (5.3), the staggered cell averages

unj+1/2,k+1/2 ≈ wn

j+1/2,k+1/2 ≡ 1

∆x∆y

∫ ∫

Cj+1/2,k+1/2

w(x, y, tn) dxdy (5.5)

=1

∆x∆y

[

∫ xj+1/2

xj

∫ yk+1/2

yk

pj,k(x, y)dxdy +

∫ xj+1/2

xj

∫ yk+1

yk+1/2

pj,k+1(x, y)dxdy

+

∫ xj+1

xj+1/2

∫ yk+1/2

yk

pj+1,k(x, y)dxdy +

∫ xj+1

xj+1/2

∫ yk+1

yk+1/2

pj+1,k+1(x, y)dxdy]

can be computed exactly. Therefore, the central scheme can be summarized through the following

tasks:

(i) Use cell averages, uj,k to reconstruct the polynomials pj,k(x, y) at time t = tn within each cell Cj,k.

(ii) Follow the evolution of the point-wise values, w(xj , yk, τ), tn ≤ tn+1, along the interfaces. For a

sufficiently small time step, these interfaces are free of discontinuities. Therefore, the fluxes in (5.4)

can be approximated to the desired accuracy by appropriate quadrature rules.


(iii) For the reconstructions in (i), one can use either a piecewise linear MUSCL interpolation or an

ENO approximation of third order or higher.

Second order NT scheme

The second order NT (Nessyahu-Tadmor) [35] scheme is obtained by the use of a piecewise linear

MUSCL interpolation leading to a second order Riemann solver-free (Lax-Friedrichs-type) method.

Thus, the approximation polynomials in (5.3) are given by

pj,k(x, y) = uj,k +Dxuj,kx− xj

∆x+Dyuj,k

y − yk

∆y(5.6)

with Dxuj,k, Dyuj,k are the MUSCL based approximate slopes in x and y directions respectively.

With the approximation in (5.6), the staggered cell averages formula in (5.5) becomes

wnj+1/2,k+1/2 =

1

4(uj,k + uj,k+1 + uj+1,k + uj+1,k+1) (5.7)

+1

16(Dxuj,k −Dxuj+1,k +Dxuj,k+1 −Dxuj+1,k+1)

+1

16(Dyuj,k −Dyuj,k+1 +Dyuj+1,k −Dyuj+1,k+1) .

The superscript n on the right-hand-side of (5.7 is omitted for simplicity in exposition.

The fluxes in (5.4) are then approximated by using a midpoint-second order quadrature rule in time

combined with a trapezoidal approximation in space, which yields

∫ tn+1

tn

∫ yk+1

yk

f(w(xj , y, τ)) dydτ ≈ ∆t∆y

2

(

fn+1/2

j,k+1+ f

n+1/2

j,k

)

. (5.8)

In order to obtain an estimate for the midpoint values fn+1/2

j,k , a predictor-corrector procedure is used.

We use a first order Taylor expansion to determine the mid-point (pointwise) values, un+1/2

j,k ,

for λ :=∆t

∆x, and µ :=

∆t

∆y, we have

un+1/2

j,k = unj,k − λ

2Dxfj,k − µ

2Dygj,k (5.9)

Dxfj,k :=df

du(un

j,k)Dxuj,k; Dygj,k :=dg

du(un

j,k)Dyuj,k


(predictor step). The ’predicted’ values in (5.8) are than injected into the integrals in (5.5) to yield a

midpoint quadrature rule. Next, the resulting second order scheme is given in a compact form.

wn+1

j+1/2,k+1/2= 〈1

2(uj,. + uj+1,.) +

1

8(Dxuj,. −Dxuj+1,.) − λ

(

fn+1/2

j+1,. − fn+1/2

j,.

)

〉k+1/2

+ 〈12(u.,k + u.,k+1) +

1

8(Dyu.,k −Dxu.,k+1) − µ

(

gn+1/2

.,k+1− g

n+1/2

.,k

)

〉j+1/2

with the staggered averages given by: (5.10)

〈φj,.〉k+1/2 =1

2(φj,k + φj,k+1); 〈φ.,k〉j+1/2 =

1

2(φj,k + φj+1,k).

Central incompressible scheme

The central incompressible scheme for the barotropic system in (3.3) is achieved by combining the

predictor-corrector central scheme formed by (5.9)-(5.10) and a solver for the Poisson equation in (3.3)

with the appropriate differentiation formulas which guaranties numerical incompressibility. We use a

five point stencil for the Poisson equation:

ψj+1,k − 2ψj,k + ψj−1,k

∆x2+ψj,k+1 − 2ψj,k + ψj,k−1

∆y2= ωj,k ≡ (ξj,k − yk) (5.11)

1 ≤ j ≤ N, 1 ≤ k ≤ M . We solve the difference equation in (5.11) by combining the FFT method

for the periodic x-direction and a direct method in y. The details of the combined Poisson solver are

presented in the Appendix I, however, it is worthwhile mentioning that to close the Poisson equation

we use the Neumann boundary condition,

∂ψ

∂x

∣

∣

∣

∣

y=±Y

= 0

dictated by the no-flow condition in (2.8), which translates immediately to the Fourier modes as a

Dirichlet boundary condition except for the mode number one where some special treatment is needed

(see the appendix for details).

Now, we construct the incompressible flow from the discrete stream-function solution provided by

the Poisson solver of (5.11). Simple second order centered finite differences, lead to

uj,k = −ψj,k+1 − ψj,k−1

2∆y; vj,k =

ψj+1,k − ψj−1,k

2∆x. (5.12)

If the second order centered differences are also used to compute the discrete divergence, then we get

Dxuj,k +Dyuj,k ≈ uj+1,k − uj−1,k

2∆x+vj,k+1 − vj,k−1

2∆x


= −ψj+1,k+1 − ψj+1,k−1 − ψj−1,k+1 + ψj−1,k−1

4∆y∆x

+ψj+1,k+1 − ψj−1,k+1 − ψj+1,k−1 + ψj−1,k−1

4∆x∆y≡ 0,

i.e., the discrete flow field, (uj,k, vj,k) obtained in this fashion is incompressible. This is conceptually

simpler and computationally cheaper than centering the incompressibility at the staggered grid points,

(j + 1/2, k+ 1/2) [26].

High order resolution

The MUSCL slopes Dxuj,k and Dyuj,k in (5.6) are computed via the nonlinear MINMOD limiter [26]

Dxuj,k = MINMOD(

θ(uj+1,k − uj,k), (uj+1,k − uj−1,k)/2, θ(uj,k − uj−1,k))

,

Dyuj,k = MINMOD(

θ(uj,k+1 − uj,k), (uj,k+1 − uj,k−1)/2, θ(uj,k − uj,k−1))

(5.13)

MINMOD(a, b, c) =

min(a, b, c) if a > 0, b > 0, c > 0

max(a, b, c) if a < 0, b < 0, c < 0

0 elsewhere

and 0 ≤ θ ≤ 2. In our numerical tests below we use values varying from θ = 1.5 to θ = 2 to minimize

the numerical dissipation.

5.2 Boundary conditions for the potential vorticity

One may argue that since the meridional component of the advecting velocity vanishes at the walls the

information outside the walls does not propagate into the domain and therefore any kind of boundary

conditions can be used there. However, this turns out to be a mistake. We demonstrate below that

the choice of suitable numerical boundary conditions for ξ at the walls is crucial. Certain types of

boundary conditions such as polynomial extrapolation develop a numerical boundary layer near the

walls where unphysical vorticity is generated and amplifies with time. We propose to use instead the

fixed Dirichlet boundary conditions for the vorticity,

ω ≡ ξ − y = 0, y = ±Y, (5.14)


which leads to clean and accurate results. At this points, the reasons for this behavior are not well

understood.

5.3 Validation with large scale Rossby wave packets

We consider the Rossby wave packets given by

ψ(x, y) = cos(k1x− ωt) sin(k2y) (5.15)

with (k1, k2) are the zonal and meridional wavenumbers, respectively, and the phase ω is given by the

dispersion relation

ω = − k1

k21 + k2

2

, (5.16)

which are exact solutions to the nonlinear barotropic equations in (3.3) (see [20]) with the boundary

conditions v = 0 at the walls if the meridional wavenumber is chosen accordingly as discussed below.

The solutions described by (5.15) and (5.16) represent a traveling wave packet which propagates in

the zonal direction (parallel to the channel walls) at the speed ω/k1. We run the barotropic code in

(5.1) through (5.13) for a fix period of time T . The (initial) magnitude of the wind is such that

maxx,y

√

u(x, y, 0)2 + v(x, y, 0)2 = 5 m/s.

We fix the meridional wavenumber k2 so that the associated wavelength matches the width of the

channel, given by 2Y = 10, 000 km, thus yielding no-flow across the walls. In the zonal direction,

we consider two different wavenumbers: k1 = 2π/X (wavenumber one, sometimes we write k1 ≡ 1)

yielding a zonal wavelength equal the total equatorial circumference, X = 40, 000 km, and k1 = 8π/X

(k1 ≡ 4) yielding a zonal wavelength identical to the meridional wavelength fixed by the channel width,

thus, resulting in a ’square wave’. Notice that the Rossby wave packet in (5.15) satisfies the Dirichlet

boundary condition in (5.14).

We report in table 3 the L1-norm error, with respect to the velocity field (u, v), between the

exact and the numerical solutions computed with two different grids, 128× 75 and 256× 150, at four

consecutive times, 5,10,15, and 20 days, for the case k1 = 2π/X with the fixed Dirichlet boundary

conditions in (5.14).



In Figure 3, we plot a zonal slice of the vorticity at roughly y = 1600 km for the exact (solid)

and the numerical solutions of table 3, obtained with the 128 × 75 (dotted) and 256 × 150 (dashed)

grids. In Figure 4 we monitor the associated barotropic kinetic energy in (2.10); the two curves named

BC1 on Figure 4. A comparison of the errors in Table 3 for the two grids suggests that the central

incompressible scheme captures the large scale dispersive wave in (5.15) with an overall first order

accuracy. Nevertheless, as demonstrated by Figures 3 and 4, this low order convergence is due to the

fact that the numerical solutions over predict the phase speed as the dotted and dashed lines in Figure

3 show up always ahead of the solid line while the energy plots of Figure 4 demonstrate at least a

second order convergence (compare the two BC1 curves).



As noted above, although the meridional flow is assumed to vanish at the walls so that the com-

putational domain does not in principle exchange information with the outside walls, the numerical

boundary conditions for the (potential) vorticity can not be arbitrarily chosen and actually constitute

a subtle issue. Numerical tests revealed some popular types of boundary conditions such as polyno-

mial extrapolations behave very poorly when compared to the fixed boundary conditions in (5.14).

The curves BC2 and BC3 in Figure 4 represent the energy-time plots obtained for the same large

scale Rossby wave when the zero order (constant) and the second order (parabolic) extrapolations to

ghost cells (outside the walls) are used, respectively. Compare with the two BC1 curves which use

the Dirichlet boundary condition in (5.14). In Figure 5 we compare the flow structure of the exact

solution (A) and the numerical solutions obtained with the three different boundary conditions listed

above for the large scale Rossby wave packet in (5.15) with zonal wavenumber one and meridional

wavenumber four at time t = 20 days. (The contours are those of the vorticity and the arrows repre-

sent the barotropic velocity profiles.) Clearly, overall, the two runs corresponding to the polynomial

extrapolation boundary conditions behave very poorly. The vorticity contours of Figure 5 (C) and (D)


exhibit a numerical boundary layer near the walls and the energy plots in Figure 4 BC2 and BC3 are

less accurate than BC1 for the same grid, especially the zero order extrapolation case shows an exces-

sive energy dissipation, almost as bad as the curve obtained by BC1 with the coarse grid of 128×75.

Furthermore, while the second order extrapolation seems to be acceptable on the energy side in Figure

4, the numerical boundary layer shown by the contours near the lower wall of Figure 5 (D), although

narrower than its counterpart in Figure 4 (C), displays an enormous unphysical vorticity amplification

(roughly by a factor five).


In order to demonstrate that the incompressible central scheme is robust and able to capture

dispersive waves with smaller wavelengths, we report in Figure 6 the results obtained with the higher

zonal wavenumber: k1 ≡ 4. Compare the flow structure, at time t = 20 days, (color contours of the

vorticity and velocity arrows) of the exact solution depicted in panel (A) with the numerical solution in

panel (B). Notice that there is no noticeable phase shift between the two panels so that the dispersion

is well captured by the numerical scheme.


6 A test on interacting barotropic and equatorial baroclinic waves

In this section we propose to test the overall code obtained by coupling the two schemes described in

sections 4 and 5 with the interaction system discretized in section 3.2 via the Strang’s fractional step

procedure. This test particularly demonstrates the robustness of the code and its usefulness for studying

the baroclinic-barotropic interactions near the equator. We use, for this purpose, an asymptotic (KdV-

like) solution, to the coupled equations in (2.3) and (2.4), which is discovered recently by Biello

and Majda [27] in a long-wave limit [20] where a large zonal wavelength compared to a meridional

wavelength of about 5400 km is assumed. This particular value for the meridional wavelength is

dictated by a resonance condition which leads to a relationship between the meridional wavenumber of


the barotropic Rossby wave and the meridional index of the baroclinic Rossby wave. The width of our

channel is thus adjusted accordingly. For simplicity, we avoid the details of derivation of the solitary

waves here. The interested reader is referred to [27]. The solitary wave of Biello and Majda is given in

the form

ψ(x, y, t) = B(x, t) sin(k2y)

u(x, y, t) = A(x, t)

(

Dm−1 −1

m+ 1Dm+1

)

v(x, y, t) =2√

2

2m+ 1

∂A(x, t)

∂xDm (6.1)

θ(x, y, t) = A(x, t)

(

Dm−1 +1

m+ 1Dm+1

)

where the amplitude functions A and B, which solve a highly nonlinear dispersive PDE system in

(x, t), are given by the formulas

A(x, t) = A+ αsech2(k1(x − cst));B(x, t) = B − βsech2(k1(x− cst))

with A and B are two closely related parameters which fix a mean barotropic wind at the equator while

α and β are two constants setting the wave amplitudes. The Dm,m = 1, 2 are the parabolic cylinder

functions [20,19]. k1 is the zonal wavenumber and k2 =√

2m+ 1 is the meridional wavenumber. In

our simulation below we pick m = 1 and cs is a negative real number, the westward traveling speed of

the solitary wave.

We run the fractional step algorithm, linking the three different routines as discussed above, with the

initial data provided by the solitary wave in (6.1). We fix the parameters so that the mean barotropic

wind at the equator blows eastward with a magnitude of 5 m/s and the zonal wavelength is on the

order of 10,000 km. The resulting (total) wave speed of the solitary wave is cs = −23 m/s. We use a

resolution of 512 X 150 grid points, yielding approximately a mesh size of 75 km in each direction x

and y, and a CFL=.9 based time step of about 15 minutes.

The history plots of the total (solid), the baroclinic (dashed), and the barotropic (dash-dotted)

energies, normalized by their corresponding initial values, up to 20 days are shown in Figure 7. In

spite of the strong interactions between the two flow components during this long-time run, reflected

by the auto-compensating ups and downs of the barotropic and baroclinic energies, the total energy


is essentially preserved as it should be the case, except for some small numerical dissipation becoming

noticeable after about t = 5 days when some small scale structures start to develop. Furthermore, this

rather small diminishing in total energy with time is expected to decrease with grid refinement.


Also from Figure 7 we can see that during the first 2 days or so the barotropic and baroclinic energies

do not change much. This actually marks the time period during which the numerical solution remain

fairly close to the asymptotic (initial) solution depicted in (6.1). In fact, in Figure 8 we display both

the baroclinic (left panels) and barotropic (right panels) flow structures by plotting the contours of the

potential temperature, θ, for the baroclinic flow and those of the vorticity, ω, with the corresponding

velocity profiles on top of them, for the asymptotic solution (E), in (6.1), and the numerical solution

(N), obtained by the coupled code, at time t = 2 days.


Up to this time, the two solutions stay fairly close to each other both qualitatively and quantitatively.

Especially, we notice that the two solutions are overall well aligned with each other, suggesting that

the phase speeds are in good agreement. However, we do see that the numerical (which provides an

approximation to the full equations; with no asymptotics) deviates from its asymptotic analogue by

showing a tilted oval shape at the center of the θ-contours on the left panel and the speed up of the

front of the off equatorial vortices shown on the right bottom panel.

Figure 9 displays the flow structures (similarly to figure 8) of the numerical solution at times

t = 5.7, 11.5, 17 days. Particularly apparent from the contours of θ, some small scale structures are

starting to form at time t = 5.7 days, also we see that the two vortices which are closer to the equator

are amplifying, thus, yielding very steep gradients. By the time t = 11.5 days, the original vortices are

split on half, thus, transferring energy into small scales.


This energy cascade becomes even more visible at time t = 17 days and the numerical solution does

no longer look like the original solitary wave. One plausible explanation for this bifurcation would be


the shear instability which is associated with the sudden increase of the barotropic energy starting

from time t ≈ 2 days. The amount of energy permitting the increase of the barotropic energy is

being extracted from the baroclinic flow (thanks to the energy conservation principle of the scheme)

thus completely destroying the initial solitary wave structure. It is worthwhile to mention here that the

barotropic mean profile utilized in the solitary wave exact solution in (6.1) is linearly unstable; however,

the asymptotic theory [27,1] allows a general, possibly linearly stable, barotropic shear, constrained

to have only the same Fourier coefficient B for ψ from (6.1). While it is not important for us here, it

would be interesting to see if such solutions are more stable in time.

7 Tests with a simple convective parametrization

7.1 The convective relaxation scheme

Now we re-couple the equation for the vertically averaged moisture with a background gradient in

(2.5) into the barotropic-baroclinic interaction equations in (2.3) and (2.4) via a simple convective

parametrization. Our goal is to test our high resolution balanced scheme on the equations with mois-

ture. Realistic convective parametrizations involve a large number of parameters and variables repre-

senting the different complex processes associated with the atmospheric convection such as the different

cloud types, condensation and evaporation, radiative effects, etc. and would also likely invoke an active

boundary layer [15,36, etc]. For the sake of simplicity, as in [24,17] we use rather a simple relaxation

scheme to parametrize the precipitation rate, P , which is set equal to the convective heating, C, in (2.7).

In order to access a certain family of exact solutions, in the form of precipitation fronts [24], which

are discussed below, all the external forces such as the sea-surface evaporation flux E, the momentum

friction, and the radiative cooling QR are also ignored here, i.e., we set E ≡ 0, cd ≡ 0, and QR ≡ 0

in (2.4). Nevertheless, it is argued in [24] that those terms do not influence much on the dynamics

of moisture fronts on which we focus here. Namely, we consider the following barotropic-baroclinic


system with a relaxation scheme convective parametrization.

(A)∂v

∂t+ v · ∇v + yv⊥ + ∇p = −1

2div (v ⊗ v)

(B) ∇ · v = 0

(C)∂v

∂t+ v · ∇v −∇θ + yv⊥ = −v · ∇v

(D)∂θ

∂t+ v · ∇θ − div (v) = P (7.1)

(E)∂q

∂t+ v · ∇q + Qdiv (v) = −P .

The precipitation rate is parametrized according to the relaxation scheme

P =1

τc(q − q(θ))+ (7.2)

where q(θ), which measures a significant factor of the saturation value of environmental moisture, sets

a moisture threshold beyond which the environment can sustain deep convection and hence produces

precipitation. (Precipitation occurs only if the moisture content, q, exceeds the ’saturation’ value,

q(θ).) In (7.2) τc is a convective characteristic time which is fixed to τc = 2 hours, see [17], [24] and

q(θ) = q + αcθ (7.3)

with αc is a positive parameter and q is a fixed positive threshold constant. A relationship between

αc and Q is obtained by imposing a fixed gravity wave speed, of 15 m/s, roughly consistent with

observations [21], within the moist regions where P > 0. This is done below. Versions of this “convective

adjustment scheme” are often used in realistic general circulation models for climate; see [17], [24] for

references.

7.2 Conservation of moist-equivalent potential temperature and moist gravity waves

When the radiative cooling and the surface evaporation are ignored, the equivalent potential temper-

ature

θe = θ + q

satisfies the equation

∂θe

∂t+ v · ∇θe − (1 − Q)divv = 0 (7.4)


i.e, θe satisfies a divergence equation, therefore it is conserved. As noted in [24], for rapid convective

adjustments (small τc’s), within the precipitation regions where P > 0, we have

q = q + αcθ +O(τc).

(Otherwise, P will explode when τc −→ 0.) Therefore, (7.4) yields a new equation for the dry potential

temperature, when τc −→ 0,

∂θ

∂t+ v · ∇θe −

1 − Q

1 + αcdivv = 0. (7.5)

By substituting (7.5) to the θ-equation in (7.1) and gathering the momentum equations (7.1)(C), a

new system of shallow water equations with gravity waves moving at a speed

cm =

√

1 − Q

1 + αc(7.6)

emerges. As in [24], we call these new waves ’moist gravity waves’. By relating cm to the speed of

observed convectively-coupled Kelvin waves (eg. [21]) which is about 15 m/s, the relationship in (7.6)

links the parameters Q and αc when the moist gravity wave speed cm is given. We use in our calculations

the value cm = 15 m/s. To close this equation and obtain plausible values for both Q and αc, we need

to impose a value for the moisture gradient Q introduced in (2.6). For a moisture gradient of Q = 1.49

g kg−1km−1, and HT = 16 km, we obtain

Q ≈ .9 =⇒ αc ≈ .025.

Still, we need to fix a value for the convective threshold, q in (7.3), for the moisture anomalies. By

looking at Fig. 9.33 in Emanuel’s book [37], this suggests the value q ≡ 2 g/kg.

7.3 Dissipation of moist energy and entropy

We introduce the moist energy

Em(t) =1

4

∫ X

0

∫ Y

−Y

(q + Qθ)2

(1 − Q)(αc + Q)dxdy. (7.7)

As discussed in Frierson, the total energy,

E = Em + Ed,


obtained by adding the moist energy in (7.7) to the dry energy introduced in (2.9), associated with

the equations in (7.1), (7.2), and (7.3), satisfies the following equation,

∂

∂tE(t) = − 1

2(αc + Q)

∫ X

0

∫ Y

−Y

(q − αcθ)P dxdy ≤ 0. (7.8)

Notice that q ≥ q + αcθ > αcθ, if P > 0 which guarantees the energy dissipation in (7.8). Therefore,

as pointed out in [24], the system in (7.1), (7.2), and (7.3), possesses a principle of energy dissipation

via precipitation. Furthermore, it is shown in [24], by similar arguments, that, in the absence of

the barotropic flow, the gradients of v, θ and q also satisfy an energy principle where an L2-like

norm involving ∇v,∇θ,∇q, namely the total energy for the gradient system, is similarly dissipated

by precipitation. Therefore, the system in (7.1) will not develop (first order) discontinuities such as

shocks during time integration. However, as we will see below, kinks in the level set contours of (

i.e, discontinuities in the first derivatives of) v, θ and q are permitted. Moreover, following [24], the

moisture system in (7.1) possesses an entropy-like function given by Z = q + Qθ, which is always

decreasing along the trajectories of the barotropic flow. In fact, combining the equations (D) and (E)

in (7.1) leads to

∂Z

∂t+ v · ∇Z = −(1− Q)P ≤ 0. (7.9)

This equation will be used below for the design a numerical scheme for the moisture system using the

different pieces of code developed in sections 3, 4, and 5.

7.4 Numerical solver for the moisture equation and its coupling into the barotropic-baroclinic

interaction code

Now we incorporate the moisture equation into the barotropic-baroclinic interaction algorithm dis-

cussed in sections 3, 4, and 5 to obtain a global solver for the equations with moisture in (7.1) with the

precipitation rate, P , given by the relaxation scheme described in (7.2) and (7.3). Since the Z-equation

in (7.9) possesses a monotonicity principle and involves less terms than the q-equation in (7.1) (E), it

is meaningful and more practical to substitute the former equation to latter in (7.1).

We expand the fractional step scheme developed above and used in section 6 to include the new

moisture equation. The forcing terms in (7.1) (D) and (7.9) are added to the interaction system in (3.4)


which is augmented with an equation for θ and an other equation for Z accounting for the dynamics

of forcing terms on the right hand side while the barotropic advection part of the Z-equation forms the

fourth piece of the new algorithm. The following two systems are substituted/added into the global

fractional step (Strang-splitting) strategy.

∂v

∂t= −1

2div (v ⊗ v) − cdv

∂v

∂t= −v · v − cdv

∂θ

∂t= P −QR (7.10)

∂Z

∂t= −(1 − Q)P +E

divv = 0

and

∂Z

∂t+ v · ∇Z = 0. (7.11)

The discretization of (7.11) combines the predictor-corrector method in time and the NT scheme of

section 5 while the algorithm for solving (7.10) remains essentially the same as described in section

3.2 except for the addition of the new forcing terms and the new equations for θ and Z. (The forcing

terms E,QR and the frictional drag coefficient cd are set to zero during the numerical tests described

below, we included them in (7.10) only for the sake of illustration.)

7.5 Limiting dynamics and precipitation fronts

In this section we introduce some special solutions for the case of a simplified version of the equations

in (7.1), reduced mainly to one space dimension with no barotropic advection, in the limit of small

convective adjustment time τc −→ 0. These solutions are constructed by Frierson et al. [24], in the

form of propagating precipitation fronts separating dry-unsaturated and saturated-precipitating re-

gions. Below we will interchangeably use the adjectives moist and dry versus the adjectives saturated

and unsaturated. We then use these limiting solutions to define initial conditions for our full two-

dimensional model in (7.1). Particularly, we investigate the beta-effect and the effect of the advection

by a barotropic trade wind-like flow on the precipitation fronts.


The detailed discussion leading to the construction of the precipitation fronts is given in [24]. Next

we give a brief summary for convenience. We consider the one-dimensional system involving only u, θ

and q.

(A)∂u

∂t− ∂θ

∂x= 0

(D)∂θ

∂t− ∂u

∂x= P (7.12)

(E)∂q

∂t+ Q

∂u

∂x= −P

with

P =1

τc(q − q − αcθ)

+.

It is established by Frierson et al. [24] that in the limit of τc −→ 0, the simplified moisture equations

in (7.12) admit three groups of simple solutions on the from of propagation fronts with jumps in

the precipitation, P , separating moist regions with P > 0 and dry regions where P = 0. They are

continuous piecewise linear solutions, (u, θ, q), with jumps in their first derivatives, (ux, θx, qx), at the

interface between moist and dry regions. In the limit of τc −→ 0, the convective system adjusts quickly

so that to leading order we have (q = q + αcθ0 within the moist regions. Therefore, in this limit, the

moist and dry regions are characterized by

Dry region: P = 0 ⇐⇒ q < q + αcθ or q = q + αcθ and∂q

∂t< 0

Moist region: P > 0 ⇐⇒ q = q + αcθ and∂q

∂t> 0. (7.13)

The exact solutions with the form of moving precipitation fronts for the the simplified system in (7.12)

at the limit τc −→ 0 are given below in terms of the jumps in the derivatives, ux, θx, qx, at the interface.

First, the Rankine-Hugoniot jump conditions are written [24]

s[w] = [θx]

s[θx] = [w] − [P ] (7.14)

s[qx] = −Q[w] + [P ]

where [f ] = f+ − f− is the jump in f at the interface located at x = 0 with f± ≡ f(0±) and w ≡ −ux

is the vertical velocity (rescaled by a factor of πLHT

) while s is the speed of the moving precipitation


front. For simplicity we assume that at time t = 0 the moist region is the entire half real line with

x > 0 while x < 0 defines the dry region. Therefore, since the precipitation is positive in the moist

region and identically zero in the dry region we have

[P ] = P+ =αc + Q

1 + αcw+, (7.15)

by using (7.12) on the moist side together with the conditions in (7.13). Notice that the positivity of

P+ implies that w+ > 0. In addition we have

q+x = αcθ+x , q

−

x ≥ αcθ−

x , and q(0) = q + αcθ(0). (7.16)

The most left equality in (7.16) follows directly from (7.13), the middle one expresses the fact that

qM − q−αcθ is a non-decreasing function of x < 0 so that the region on the left of the interface remains

dry, and the most right condition follows from a continuity argument. Combining the jump conditions

in (7.14) with the constraints in (7.15) and (7.16) leads to the following set of solutions

s = ±(

1 − αc + Q

1 + αc

w+

[w]

)1/2

[θx] = s[w] (7.17)

[qx] =

(

1 − Q

s− s

)

[w].

The equations in (7.17) form a parameter family of solutions to the convective system in (7.12) under

the constraints in (7.15) and (7.16). Furthermore, in order to get a real valued propagation speed, one

of the following conditions is required

w− ≤(

1 − αc + Q

αc + 1

)

w+ or w− ≥ w+ > 0.

Therefore, according to Frierson et al. [24], the following three groups of precipitation fronts are

possible.

– Drying fronts with the interface moving toward the moist region at a speed s such that

cm ≤ s ≤ 1

where cm is the moist gravity wave speed introduced above. (Recall that the dry gravity wave speed

is the unity here.)


– Slow moistening fronts with the interface moving toward the dry region at a speed satisfying

−cm ≤ s ≤ 0.

– Fast moistening fronts with speeds satisfying

s ≤ −1.

Although only the drying fronts satisfy the Lax stability criterion for moving fronts [38,39,24], some-

what surprisingly all the three groups of precipitation fronts are successfully demonstrated, by numer-

ical simulations in [24], to persist for small but reasonably finite adjustment times τc of a few hours.

In our simulations below we pick τc = 2 hours as utilized for example in [17].

7.6 Dying moisture wave

Our setup for numerical simulation with the equations with moisture in (7.1) consists on the design

of a dying moisture wave by the aid of the precipitations fronts of Frierson et al. [24] discussed above.

Recall our computational channel-domain covers the equatorial belt [0, X ] × [−Y, Y ] where X is the

perimeter of the earth at the equator and Y is the distance of the channel walls from the equator. We

divide this domain perpendicular to the equator into three regions: two dry regions on the zonal edges

of the domain and a moist region in the middle. We then have two moist-dry interfaces. We design

a dying-moisture-wave structure by inserting a slow moistening front at the right interface making

the moist region move toward the dry region on the right and a drying front on the left interface to

move the other dry region toward the moist region in the middle, see Figure 10. Notice that the two

interfaces are moving in the same direction, to the right, hence the whole structure is moving in that

direction but since the drying front is moving faster than the slow moistening front we expect the

moist region to die out in a finite time. This why we call it a dying moisture wave.

By imposing the constraints listed in the previous section on both interfaces a detailed algorithm

for the construction of the dying-moisture-wave is given in Appendix II. Particularly, we have the

same-identical solution within the moist region satisfying the constraints imposed on both interfaces.

The initial width of the moist region is 10,000 km, i.e., one fourth of the total domain width. Next, we


discuss the results obtained by the moisture-scheme using the dying moisture wave as initial condition

for the following cases. First, we start with the simplest case with no rotation and no barotropic

advection, where we investigate both the effect of a finite but small τc on the precipitation fronts

and their interactions with dry gravity waves. Then we gradually add the effect of the advection by a

barotropic trade wind, the beta effect, and finally both the advection and the rotation together.

7.7 Validation with τc = 2 hours and interaction with dry gravity waves

Here we run the simplified one-dimensional system in (7.12) with the initial condition in Figure 10.

(Actually, we have a two-dimensional flow in the x-z plane.) Our aim is to investigate the effects of

a finite adjustment time for the typical value of τc = 2 hours on the dying-moisture-wave which is

in theory a solution at the limit τc −→ 0. In figure 11, we plot the contours of the zonal velocity u

together with the color filled contours of the precipitation rate, P , on top of them in the x-t plane

(longitude-time). The dashed lines represent the position of the precipitation fronts associated with

the drying and slow-moistening fronts as predicted by the limiting analysis.



The agreement between the asymptotic theory and the numerics with the finite τc = 2 hours is

well established, in Figure 11, and the scenario of the dying-moisture-wave is reproduced with a good

accuracy. However, in order to achieve this, we had to give up the natural periodic boundary conditions

in x and use instead a linear extrapolation expanding the limiting solution beyond the boundaries.

Indeed, periodic boundary conditions at x = 0, X introduce a jump in the initial conditions from which

two opposite dry gravity waves emerge. It turns out that the interaction of the dry gravity waves with

the moisture fronts lead to interesting dynamics of wave interaction and reflection for the precipitation

fronts.

As depicted in Figure 12, the eastward gravity wave emanating from x = 0 at t = 0 (following the

cusps in the zonal velocity contours on the left of the color-filled precipitation region, a rarefaction


wave), which is weaker but closer to the precipitation wave, first hits the drying front at around time

t = 3.5 days. At this time, a fast moistening front is reflected and propagates backward to the west at

speed larger than 50 m/s (compare to the shock wave on the right which propagates at 50 m/s; this

is a fast moistening front). The second propagating gravity wave emanating from x = X at t = 0 is

characterized by the steep gradient region (called “a shock wave”) is propagating to the west toward

the moving precipitation area. When the shock wave hits the moist region no fronts are reflected,

instead it penetrates inside (like a dry intrusion) and dries out the whole structure while the reflected

fast moistening front dies out after having traveled a distance of about 10,000 km. This last feature is

perhaps due to the dissipation of energy by precipitation in (7.8).


7.8 Effect of the advection by a barotropic trade wind

Now, we add the trade wind-like barotropic flow, v = (u, v),

u = −u0 cos

(

3πy

Y

)

, v = 0 (7.18)

with u0 = 5 m/s, depicted in Figure 13, to the equations in (7.1) (C), (D) and (E). We ignore the

rotation effect, β = 0, the barotropic dynamics governed by (A) and (B), and the baroclinic-barotropic

interaction terms on the right hand side of (C). We set the initial conditions to the dying moisture

wave in x modulated by the cosine function in y to avoid a jump at the boundaries y = ±Y . A

truncation of the data by Dirichlet boundary conditions at y = ±Y , would induce other dry gravity

waves traveling in the north-south direction and may render the simulation very complex. We keep the

the periodic boundary conditions in x in spite of the dry gravity waves emanating from the induced

discontinuity, because otherwise (with the linear boundary conditions in x) the barotropic advection

induces overwhelming growth which causes blowup.



The initial condition for this test is therefore given by

u(x, y, 0) = cos( πy

2Y

)

ul(x)

v(x, y, 0) = 0 (7.19)

θ(x, y, 0) = cos( πy

2Y

)

θl(x)

q(x, y, 0) = cos( πy

2Y

)

ql(x)

where ul, θl, ql is the piecewise linear limiting solution representing the dying moisture wave in Figure

10.

The histories of both the total (dry + moist) energy in (7.7) and the precipitation rate are plotted

in Figure 14 (A). Perhaps, the most interesting thing we notice from Figure 14 (A) is that, unlike

the case without barotropic advection in Figure 12, the precipitation survives beyond the 9 days time

period. Notice the significant dissipation of total energy which ceases around t = 13 days when the

precipitation becomes zero. This indeed confirms the dissipation of energy by precipitation as stated

above. The longitude-latitude (x-y) contours of the water content, q, and the color filled contours of

the precipitation rate, P , at times t = 0, 1, 3, 6, 9, 13 days are plotted in Figure 15. Already at time

t = 1 day, at some distance away from the equator where the barotropic wind blows eastward, i.e, in

the same direction as the moving moisture wave, the precipitation disappears. It becomes concentrated

at the equator, which explains the first increase of the precipitation maximum on Figure 14 (A).


As the precipitation front, confined to the equator, is trying to make its way forward in the eastward

direction, as it would do in the absence of the barotropic advection, the trade wind, which blows

westward at the equator, is pushing it backward to the west. This double effect stretches the moist

region and expands it from 10,000 km to about 20,000 km wide at time t = 1 day. The effect of the

eastward (rarefaction) wave emanating from the boundary conditions and moving eastward, as noted

above, is suspected to be responsible for the formulation of the bowling pin-like shape located at time

t = 1 day, around x = 4300 km (which is the distance traveled in 1 day at a speed of 50 m/s), probably,

caused by the reflection of a fast moistening front in a scenario similar to section 7.7 above.


Notice also that at around time t = 1 day the precipitation rate attains a maximum peak of close to

8 K/day as depicted in Figure 14 (A). Not much is happening between time t = 1 day and time t = 3

days, except for poleward spread out of the precipitation while its maximum decreases. Nevertheless,

we see the shock wave emanating from x = X , located between x = 25, 000 km and x = 30, 000 km

at time t = 3 days, approaching toward the moist region. Notice the local peak in the precipitation

maximum plot, in Figure 14 (A), just before time t = 6 days. This is suspected to coincide with the

arrival of the dry-shock wave. When the shock hits the moist region, it splits it into two patches of

’active convection’, as shown in Figure 15 at time t = 6 days. Later on the two pieces merge together to

form again a single convection patch at time t = 9 days and continues to decay until the precipitation

dissipates away and disappears at about time t = 13 days.


7.9 Beta-effect on moisture fronts

We now activate the Coriolis force but switch off the barotropic advection, i.e, we consider the equations

in (7.1) (C), (D) and (E) with v = 0. The corresponding time histories of energy and precipitation

are plotted in Figure 14 (B). We see that things have changed and become more complicated. The

precipitation maximum displays peaks, that sometimes exceed 10 K/day, with interference by troughs

which are sometimes associated with dry episodes of up to 1 day. The time scale of these oscillations

varies from about 8 to 12 hours, i.e, longer than the convective characteristic time τc = 2 hours.

In Figure 16, we display a few successive snapshots of the flow moisture by drawing the contours

of the vertically averaged water content, q, and the color-filled contours of the precipitation rate, P ,

on top of them. Clearly, the moisture dynamics are much more complex than in the case without

rotation. Perhaps, the most striking phenomenon is the way in which the moisture region deforms and

disappears under the beta effect and more excitingly the way in which small islands of precipitation

form and merge together. First, we see that the precipitation region, which is initially distributed

across the whole channel, quickly shrinks to become a narrow strip centered at the equator as depicted


in Figure 16 at time t = 1 hour. Under the effect of rotation the strip elongates poleward while its

width diminishes tending to acquire a roundish shape, as shown at time t = 7 hours.



This rounding in shape happens while the precipitation maximum increases as suggested by the

plot in Figure 14 (B). Then suddenly, at time t = 14 hours, two small islands starts to form at

some distance symmetrically away from the equator. The small islands grow quickly resulting in three

patches of precipitation which merge together afterward, at time t = 20 hours, to form again a single

roundish region which is somewhat located ahead of the one at time t = 14 hours. As its front is

stretched eastward at the equator, as seen at time t = 29 hours, it suddenly shrinks backward like a

falling spring hitting the ground. This is perhaps due to some effect of the shock wave formed by the

boundary condition at x = X and propagating at the dry gravity wave speed toward the moisture

wave, reflecting back fast inertia-gravity waves, which are now present in the system with rotation.

The structure then becomes a thin stripe oriented north-south as seen at time t = 39 hours which then

splits onto two pieces and disappears. At time t = 2 days we record the first dry episode which lasts

about 6 hours until two islands of precipitation form at some distance off the equator. Interactions with

equatorial waves of all kind are suspected to be the main mechanisms underlying the rich dynamics of

the precipitation regions.

The dynamics continue under the different scenarios involving formation of small islands of precip-

itation, growing, merging, deformation, splitting, and finally dissipation. However, at later times the

picture seems to be more complex with the co-existence of several islands at once, especially after the

dry (shock) wave passes through the moist region and gets smoothed away, as shown on Figure 16 at

time t = 10 days.


7.10 Beta-effect and barotropic advection simultaneously

Now we put together the effect of rotation and barotropic advection and look at their effect on the

precipitation fronts. We run the full coupled system in (7.1) with the initial conditions given by (7.19)

for u, v, θ, q while the barotropic flow at time t = 0 is given by the trade wind-like barotropic jet in

(7.18) and Figure 13. Notice that both the barotropic code for the system in (3.3) and the baroclinic-

barotropic interaction system are activated so that the barotropic shear also undergoes dynamical

changes. In Figure 14 (C) we plot the history of the total energy and of the precipitation maximum

associated with this last experiment.

We see clearly that if the moisture dynamics are not more complex than the case with rotation

alone, they are at least comparable. Indeed, in Figures 17 and 18, we plot the structure of the resulting

flow at an early time of 20 hours and at a later time of 10 days during the simulation. We see clearly

that the moisture plots on the bottom resemble their analogues in Figure 16, for the case with the beta

effect alone, except for some displacement and stretching due to the barotropic flow as experienced

above for the case with advection alone. The two top panels on Figures 17 and 18 display the barotropic

(topest panels) and baroclinic (middle panels) flows on top of the vorticity and potential temperature

contours, respectively. As the time grows we see that the barotropic trade wind undergoes interesting

changes resulting from its interaction with the baroclinic flow through energy exchange. This is clearly

demonstrated by the plots in Figure 14 (D) which represent the time histories of the total dry energy

Ed in 2.11) (solid line) and the baroclinic (dash line) and barotropic (dash-dot line) contributions.

While the baroclinic and barotropic curves undergo violent and mutually compensating oscillations,

their sum (the total dry energy) is relatively smoothly increasing suggesting an extraction of energy

from precipitation (the moist energy). Also notice the overall increase in barotropic energy. Indeed,

from Figures 17 and 18, the barotropic wind becomes significantly stronger at the equator, increasing

to roughly twice its initial amplitudes, and small scale vortices emerge. The baroclinic flow on the other

hand becomes trapped at the equator where a zone of strong convergence appears particularly at and

near the precipitation patches. The fluid is strongly sucked toward the equator, hence suggesting a

local Hadley-like circulation.




8 Concluding discussion

A high resolution geostrophically balanced scheme has been designed for a simplified climate model

by using state of the art numerical methods. The model has a crude vertical resolution reduced to a

barotropic and a first baroclinic modes. Nevertheless, it acknowledged that such idealized models are

widely and successfully used for the studies of the interactions between midlatitude and equatorial

Rossby waves as well as they form an useful tool for the validation and the assessment of convective

parametrizations. The high resolution balanced numerical model is suggested as a complementary al-

ternative to the existing balanced asymptotic models such as the WTG and the multiscale models. The

barotropic/baroclinic model consists on a highly nonlinear interacting system which conserves total

energy in the absence of convective heating and frictional forcing and when suitable boundary condi-

tions are used. We consider a zonally periodic channel centered at the equator with no-flow boundary

conditions at the longitudinal walls. We adapt a basic discretization strategy consisting of fractional

operator splitting of the interacting system into three different subsystems which independently con-

serve energy: A shallow water system for the advected baroclinic waves, an incompressible barotropic

system, and a third system gathering the nonlinear interaction terms.

The advection is present in both the baroclinic and the barotropic subsystems but, however, as a

consequence of the Galerkin projection, it is restricted to the advection by the incompressible barotropic

flow, therefore, yielding a (linear) baroclinic subsystem which can be writing as a system of conservation

laws with source terms; the Coriolis terms. We use the f-wave algorithm, proposed by Bale et al. [25] for

conservation laws with varying flux functions and source terms, to discretize the advected baroclinic

waves system. The validation tests, in section 4, using known exact solutions among the large scale

equatorial waves, such as the Kelvin, the Yanai, and the Rossby waves of index one and two, have

confirmed the second order accuracy of the balanced scheme, as demonstrated by the L1-norm errors

in Table 1 and the energy plots in Figure 1. The judiciously chosen validation tests among which we


have non-dispersive (Kelvin) and dispersive (Yanai and Rossby) as well as geostrophically balanced

(Kelvin), weakly balanced (Rossby), and unbalanced (Yanai) waves, form a representative set of the

complex tropical wave dynamics.

We also looked at the effect of the boundary conditions for a flow confined into a channel, such as

the non-reflecting and the homogeneous Dirichlet boundary conditions at the channel walls, on these

natural exact solutions of the equatorial beta-plane shallow water equations. It turns out that with

the physical equator-wall distance of Y = 5000 km, for the range of grid mesh sizes varying from

about 300 down to 70 km in x and from about 130 down to 30 km in y, the L1-norm between the

exact solution and the numerical solution for the large scale MI=1 Rossby wave with the artificial

boundary conditions listed above, although it remains very small, it doesn’t decrease significantly

with grid refinement, as shown in Table 2. Nevertheless, as demonstrated in Figure 2 (A), these small

errors are confined near the channel walls where the artificial boundary conditions were imposed. More

importantly, as shown in Figure 2 (B), the energy is essentially conserved as the numerical dissipation

is negligibly small and seem to be controlled (behave well) under grid refinement. A more conservative

wall distance of Y = 8000 km permits to recover the second order accuracy in both the L1-norm

errors in Table 2 and the energy plots in Figure 2 (D). The error distribution plot, for this case with

Y = 8000, showing an overall error, with the finest grid, concentrated away from the walls as a result

of the trapping effect, provides and explanation to why the second order accuracy is recovered and

suggests that the channel geometry with a sufficiently large wall distance with either the non-reflecting

or the Dirichlet boundary conditions yields an accurate numerical model for the interactions of the

equatorially trapped waves of all nature.

It is demonstrated here that the non-oscillatory central scheme, introduced by Levy and Tadmor

for the incompressible Euler equations, constitutes an accurate computational tool for the dispersive-

barotropic Rossby waves in a channel. The validation tests of section 5, using Rossby wave packets as

exact solutions have confirmed this. The L1-norm errors displayed in Table 3 demonstrates that we

have roughly a first order convergence with grid refinement but, nevertheless, the solid and dashed

curves among the energy plots in Figure 4 show that we have at least a second order convergence energy


wise. The explanation for this discrepancy, in the order of convergence, comes from the vorticity slices

shown in Figure 3, which demonstrates clearly that the numerical scheme over-predicts the phase

speed of the wave packet as we see the dashed and dotted lines always show up at the front. It is

also demonstrated that the naive use of polynomial extrapolation leads to catastrophic generation

and amplification of vorticity within a numerical boundary layer near the channel walls as well as an

excessive dissipation of energy, at least for the case with zero order extrapolation.

Both the strategy of fractional splitting into three major pieces which independently conserve energy

and the judicious choice of the appropriate high resolution and high order methods to solve each one

of the subsystems have contributed to the success of the numerical test on interacting barotropic

baroclinic waves consisting on the solitary wave KdV solution of Biello and Majda [27] of section 7.

It is particularly demonstrated that the numerical scheme developed here was able to carry on the

violent interactions between the baroclinic and barotropic components of the flow with no significant

ad hoc dissipation of total energy. This is happening in spite of the apparent formation of small scale

structures which are perhaps due to the development of a shear instability after the first few days of

evolution.

Finally, the tests performed in section 8 with a simple convective parametrization, using the pre-

cipitation fronts limiting exact solutions from [24], which are demonstrated to persist for the typical

convective time of τc = 2 hours in Figure 11, have demonstrated the usefulness of such a numerical

scheme for tropical climate studies through the interactions of equatorial waves with both the mid-

latitudes and cumulus convection. The preliminary runs shown in section 8, already, show how rich

and complex are the interactions of convection with tropical waves. First we demonstrated that the

interaction of the moisture front with a dry gravity wave can either lead to the reflection of a new

precipitation front or the complete extinction of the precipitation, depending on the structure and

strength of the wave, as seen in Figure 12. Unlike the case with advection alone, we saw that the beta

effect can make patches of precipitation appear at new places, merge together to form larger regions

of saturated air, and disappear, as suggested by Figures 14, 15 and 16. This is perhaps due to the

capacity of the beta-plane system to support a large diversity of waves. We saw also that the effect


of the advection by a barotropic trade wind-like flow is mainly to stretch and displace the moisture

patches. Furthermore, the combination of both the barotropic advection and the beta effect is also an

interesting experiment. While the complex dynamics of the precipitation patches induced by the beta

effect remained as in the case with rotation alone, the trade wind interacts with the baroclinic moist

dynamics and becomes twice stronger through energy exchange at the equator and small scale vortices

are clearly seen at time t = 10 days (the top panel of Figure 18). The other noticeable interesting

feature comes from the middle panel of Figures 17 and 18. We see clearly a strong convergent flow

toward the equator suggesting a local Hadley like circulation, where the air rises at the equator and

descends at high latitudes.

Acknowledgments

The research of B.K. is supported by a University of Victoria Start-up grant and a grant from the

Natural Sciences and Engineering Research Council of Canada. The research of A.M. is partially

supported by ONR N0014-96-1-0043, NSF DMS-96225795, and NSF-FRG DMS-0139918.

The use of the f-wave algorithm in this paper was motivated after a fruitful discussion between the

first author and LeVeque when he was visiting UVic. The authors are thankful to J. Biello for his help

in providing the detailed formulas for the solitary waves used in section 6.

Appendix A: Poisson solver for incompressible flows in a channel

Here we discuss the numerical Poisson solver used for the computation of the incompressible barotropic

velocity field from the vorticity. Given the vorticity

ω = ξ − y


where ξ is the potential vorticity obtained by the evolution scheme, we seek the stream function, ψ,

and the incompressible flow field, (u, v), solution of

∆ψ = ω in [0, X ]× [−Y, Y ] (A1)

u = −∂ψ∂y

; v =∂ψ

∂x

ψ(X, y) = ψ(0, y), y ∈ [−Y, Y ] (periodicity)

v(x,−Y ) = v(x, Y ) = 0 x ∈ [0, X ] (no-flow).

Since we have periodicity in x, we take advantage of the availability of Fast Fourier Transforms and use

a Fourier method to discretize in that direction. Namely, we use the sine and cosine expansion which

is more efficient and more suitable for real valued functions. Assume that the computational domain

[0, X ] × [−Y, Y ] is uniformly divided onto N ×M number of grid cells. We denote the grid points by

(xj , yk): xj = (j−1)∆x, 1 ≤ j ≤ N and yk = −Y +(k−1)∆y, 1 ≤ k ≤M where∆x =X

N,∆y =

2Y

M − 1

are the mesh sizes. The sine and cosine expansion of the resulting discrete stream-function,

ψj,k ≈ ψ(xj , yk), 1 ≤ j ≤ N, 1 ≤ k ≤M,

in the periodic-x-direction is given by

ψj,k =N∑

l=1

rl,kφl,j

with (A2)

φ2p−1,j = cos

(

2π(p− 1)(j − 1)

N

)

φ2p,j = sin

(

2π(p− 1)(j − 1)

N

)

where

1 ≤ p ≤ N2

if N is even

1 ≤ p ≤ N+1

2if N is odd

where rl,k represents the Fourier mode # l associated with the grid point yk. Notice that in particular

φ1,k ≡ 1, therefore, r1,k =1

N

N∑

j=1

ψj,k.

The five-point discretization for the Laplacian in (A1) leads to the difference equation

ψj+1,k − 2ψj,k + ψj−1,k

(∆x)2+ψj,k+1 − 2ψj,k + ψj,k−1

(∆y)2= ωj,k, 1 ≤ j ≤ N, 1 ≤ k ≤M (A3)


Inserting the expansion in (A2) into (A3) leads to N M ×M tridiagonal matrix systems for the N

Fourier modes

(rl,k+1 + βlrl,k + rl,k−1) = (∆y)2ωl,k. (A4)

Here ωl,k denotes the discrete Fourier Transform of ωj,k. The coefficients, βl, in (A4) are given by

βl = −2

(

1 + 2

(

∆y

∆x

)2

sin2

(

π[l/2]

N

)

)

(A5)

where [l/2] represents the largest nonnegative integer which is smaller that l/2.

The systems in (A4) are solved (directly and efficiently) by the help of the boundary conditions at

y = ±Y in (A1), expect for the first Fourier mode where a more sophisticated argument is used to

close the associated system. Indeed, the finite difference version of the condition

ψx = 0 at y = ±Y

implies the vanishing of rl,k at the boundaries y1 = ±Y , provided l ≥ 2. Thus, except for the first

mode (l = 1), the systems in (A4) are augmented with the Dirichlet boundary condition

rl,1 = rl,M = 0. (A6)

To close the system in (A4) for the first Fourier mode we rely to the conservation of the zonal mean

of the zonal barotropic velocity u at the walls. Given the initial zonal flow, u0, we have for both the

incompressible system for the barotropic flow ( in (3.2)) and interaction system in (3.4), the zonal

mean of the zonal velocity

〈u〉 =1

X

∫ X

0

u(x, y)dx

is conserved at the boundaries y = −Y, Y . The conservation of mean zonal wind at the walls for the

two system is easily obtained by taking the zonal averaging of the first momentum equation in (2.3)

and making use of the no-flow condition in (2.8) for both the baroclinic and barotropic flows. This

conservation condition translates to the Poisson equation in (A1) as a Neumann boundary condition

for the first Fourier mode in (A4)

r1,2 − r1,0 = 2∆y〈u0〉|y=−Y (A7)

r1,M+1 − r1,M−1 = 2∆y〈u0〉|y=Y .


The boundary conditions in (A6) and (A7) are used to solve the systems in (A4) and enforced at the

ghost cells, k = 0, k = M + 1, on the Fourier modes before performing the inverse Fourier transforms

to retrieve the stream-function ψj,k at the grid points 1 ≤ j ≤ N, 0 ≤ k ≤ M + 1 including the

ghost points, k = 0 and k = M + 1, necessary for the finite difference construction of the discrete

incompressible flow, uj,k, vj,k, 1 ≤ j ≤ N, 1 ≤ k ≤M in (5.12).

Appendix B: Algorithm for a two fronts-moisture wave

Here we give the details of the algorithm used to generate the initial data for the moisture runs. We

use the index i = 1, 2, 3 to denote from left to right the three different regions in Figure 10. Let

ui = uixx+ ci, θi = θi

xx+ di, qi = qixx+ ei (B1)

where the slopes X ix are such that

X1x = Xx(a−), X2

x = Xx(a+) = Xx(b−), X3x = Xx(b+)

where X represents u, θ, or q with ux = −w. Suppose that the interface on the left is at x = a and

the one on right is at x = b. Given the parameters Q, αc, and q as in the text, we have following

Algorithm:

1. Fix the speed, sa, of drying front (on the left of moist region) such that cm ≤ sa ≤ cd = 1

2. Fix the vertical velocity within the moist region w(a+) = w(b−)

3. The vertical velocity in the left dry region is

w(a−) =

(

1 − K

1 − s2a

)

w(a+),K =αc + Q

1 + αc

4. Fix the speed, sb, of moistening front (on the right of moist region) such that 0 ≤ sb ≤ cm

5. The vertical velocity in the right dry region is

w(b+) =

(

1 − K

1 − s2b

)

w(a+),K =αc + Q

1 + αc

6. Fix θx(a+) = θx(b−), slope of pot. temp. within the moist region


7. The slope of moisture in the moist region satisfies

qx(b−) = qx(a+) = αcθx(a+)

8. The rest of the slopes follow

θx(a−) = θx(a+) − sa(w(a+) − w(a−)), θx(b+) = θx(a+) + sb(w(b+) − w(a+)),

qx(a−) = qx(a+) −(

1 − Q

sa− sa

)

(w(a+) − w(a−)),

qx(b+) = qx(a+) +

(

1 − Q

sb− sb

)

(w(b+) − w(a+))


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25. D. Bale, R. LeVeque, S. Mitran, and J. Rossmanith. A wave propagation method for conservation laws and balance

laws with spatially varying flux functions. SIAM Jour. Sci. Comput., 24:955–978, 2002.

26. D. Levy and E. Tadmor. Non-oscillatory central schemes for the incompressible 2-D Euler equations. Mathematical

Research Letters, 4:1–20, 1997.

27. J. Biello and A. Majda. The effect of meridional and vertical shear on the interaction of equatorial baroclinic and

barotropic rossby waves. Studies in App. Math., 112:341–390, 2003.

28. R. Kupferman and E. Tadmor. A fast, high resolution, second-order central scheme for incompressible flows. Proc.

Nat. Acad. Sci., 94:4848–4852, 1997.

29. R. LeVeque. Balancing source terms and flux gradients in high-order Godunov methods: The quasi-steady wave-

propagation algorithm. J. Comput. Phys, 146:346–365, 1998.

30. E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with

hydrostatic reconstruction for shallow water flows. SIAM Jour. Sci. Comput.’, 25:2050–2065, 2004.

31. F. Bouchut, J. Le Sommer, and V. Zeitlin. Frontal geostrophic adjustment and nonlinear wave phenomena in one

dim rotating shallow water: part 2: high resolution numerical simulations. J. Fluid Mech, in press, 2004.

32. R. LeVeque. Numerical methods for conservation laws. Birkhauster, Basel, 1990.

33. D. Durran. Numerical methods for wave equations in geophesical fluid dynamics. Springer-Verlag, NY, 1999.

34. L. Evans. Partial differential equations, Graduate Studies in Mathematics, Vol. 19. Amer. Math. Soc., 1998.

35. Z.-C. Zhang and S. John Yu. A non oscillatory central scheme for conservation laws. In AIAA Fluid Dynamics

Conference, 30th, Norfolk, VA, June 28-July 1, AIAA-1999-3576, 1999.


36. J. Zehnder. A comparison of convergence- and surface-flux-based convective parametrizations with applications to

tropical cyclogenesis. J. Atmos. Sci., 58:283–301, 2001.

37. K. Emanuel. Atmospheric convection. Oxford University Press, 1994.

38. P. Lax. Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math., 10:537–566, 1957.

39. A. Majda. The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. Vol. 41. Amer. Math. Soc,

1983.


List of Figures

1 Baroclinic energy (normalized by the total domain area) versus time. Computation ofMI=2 Rossby wave with the f-wave algorithm. Exact boundary conditions at the zonalwalls are used. MC limiter with CFL=.9. . . . . . . . . . . . . . . . . . . . . . . . . . 61

2 Role of boundary conditions and distance from walls on the approximation of equatorialwaves. Case of MI=1 Rossby wave with boundary condition BC#2. Error distributionon the finest grid of 512 × 300: (A) Y = 5000 km, (C) Y = 8000 km, and energy-timeplots with three different grids: (B) Y = 5000 km, (D) Y = 8000 km. . . . . . . . . . 62

3 Zonal slice plots of the vorticity, ω, at y ≈ 1600 km and t = 5, 10, 15, 20 days forexact (thick solid) and the numerical solutions. Large scale Rossby wave packet withk1 ≈ 1, k2 ≈ 4. 128×75 (dotted) and 256× 150 (thin solid) grid points. . . . . . . . . 63

4 Barotropic energy-time plots obtained by the incompressible central scheme with thefixed Dirichlet boundary conditions (BC1) using 128×75 and 256× 150 grid points andzero order extrapolation (BC2) and second extrapolation (BC3) on a 256×150 grid.Case of a Rossby wave packet with k1 ≡ 1 and k2 ≡ 4. . . . . . . . . . . . . . . . . . . 64

5 2d structure of the flow for the Rossby wave packet with k1 ≡ 1 and k2 ≡ 4 at timet = 20 days. Color contour of the vorticity and velocity profile (arrows). exact solution(A), and central incompressible scheme with fixed Dirichlet (B), zero order extrapolation(C), and second order extrapolation (D) boundary conditions at the walls. . . . . . . . 65

6 Same as in Figure 5 but with a zonal wavenumber k1 ≡ 4. Exact solution (A) andnumerical solution with Dirichlet boundary conditions (B). . . . . . . . . . . . . . . . 66

7 Time history of the total (solid), the baroclinic (dashed), and the barotropic (dash-dotted) energies, normalized by their corresponding initial values, for the barotropic-baroclinic interacting solitary wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Baroclinic and barotropic structure of the solitary wave at time t = 2 days. Comparisonof the KdV-asymptotic solution (top panels) and the numerical solution (bottom panels).Contours of the potential temperature and those of the vorticity, respectively, on top ofthe corresponding velocity profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9 Same as Figure 8 but for the numerical solution at times t = 5.7, 11.5, 17 days. . . . . 69

10 Sketch of the dying moisture wave: two interfaces separating the dry regions and themoist (shaded) region. They are moving in the same direction with different speeds (slowmoistening and drying fronts). The thick arrows represent the direction and magnitude,s, of the front speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

11 Validation of the dying moisture wave with τc = 2 hours, linear extrapolation boundaryconditions at x = 0, X . Contours of zonal velocity, u, color-filled contours of precipita-tion, and position of the drying and slow-moistening fronts as predicted by the theoryfrom [24] (dash and dash-dot lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

12 Same as Figure 11 but with periodic boundary conditions at x = 0, X . An eastward(rarefaction) and a westward (shock) dry gravity waves are created by the jump atx = 0, X caused by the periodic boundary conditions. . . . . . . . . . . . . . . . . . . 72

13 Barotropic trade wind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

14 History of total (dry+moist) energy (dashed) and precipitation rate (solid) for the dyingmoisture wave experiment: (A) with a trade wind barotropic advection, (B) with betaeffect, and (C) with both barotropic advection and beta effect. The panel (D) displaythe time histories of the total dry energy (solid) and the baroclinic (dash) and barotropic(dash-dot) contributions for the case in (C). . . . . . . . . . . . . . . . . . . . . . . . . 74

15 Contours of the water content, q, (g/kg) and precipitation rate, P , (color filled, K/day)for the dying moisture wave experiment with trade wind barotropic advection. Timest = 0, t = 1 day, t = 3 days, t = 6 days, t = 9 days, and t = 13 days. . . . . . . . . . . 75

16 Same as in Figure 15 but for the experiment with the beta effect at times 1 hour, 7hours, 14 hours, 20 hours, 1 day, 1 day and 5 hours, 1 day and 15 hours, 2 days, 2 daysand 6 hours, and 10 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


17 Profile of the barotropic (top) and the baroclinic (middle) flows on top of the contours ofvorticity and potential temperature, respectively, and contours of the vertically averagedmoisture content, q, on top of , color filled contours of the precipitation rate (bottom).Experiment with both beta effect and barotropic advection (trade wind initially). Timet = 20 hours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

18 Same as Figure 17 but at time t = 10 days. . . . . . . . . . . . . . . . . . . . . . . . . 79

FIGURES 61

0 5 10 15 20 25 30 35 40 45 50635.8

635.85

635.9

635.95

636

636.05

636.1

636.15

636.2

636.25

636.3

time (days)

EN

ER

GY

(J/

kg)

128 × 75256 × 150512 × 300

Fig. 1. Baroclinic energy (normalized by the total domain area) versus time. Computation of MI=2 Rossby wave withthe f-wave algorithm. Exact boundary conditions at the zonal walls are used. MC limiter with CFL=.9.

62 FIGURES

0 5 10 15 20 25 30103.1

103.11

103.12

103.13

103.14

103.15

103.16

103.17

Time (days)

Ene

rgy

(J/k

g)

(B) 128× 75256× 150512× 300

0 5 10 15 20 25 3064.445

64.45

64.455

64.46

64.465

64.47

64.475

64.48

64.485

64.49

64.495

Time (days)

Ene

rgy

(J/k

g)

(D)128× 75256× 150512× 300

Fig. 2. Role of boundary conditions and distance from walls on the approximation of equatorial waves. Case of MI=1Rossby wave with boundary condition BC#2. Error distribution on the finest grid of 512 × 300: (A) Y = 5000 km, (C)Y = 8000 km, and energy-time plots with three different grids: (B) Y = 5000 km, (D) Y = 8000 km.

FIGURES 63

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

0.1

ω @

y=

1600

km

5 days

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

0.1

10 days

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

0.1

15 days

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

0.1

20 days

X (1000 km)

ω @

y=

1600

km Exact

256 × 150128 × 75

Fig. 3. Zonal slice plots of the vorticity, ω, at y ≈ 1600 km and t = 5, 10, 15, 20 days for exact (thick solid) and thenumerical solutions. Large scale Rossby wave packet with k1 ≈ 1, k2 ≈ 4. 128×75 (dotted) and 256 × 150 (thin solid)grid points.

64 FIGURES

0 5 10 15 20 252.9

2.95

3

3.05

3.1

3.15

Time (days)

Ene

rgy

(J/k

g)

BC1: 256 × 150BC1: 128 × 75BC3: 256 × 150BC2: 256 × 150

Fig. 4. Barotropic energy-time plots obtained by the incompressible central scheme with the fixed Dirichlet boundaryconditions (BC1) using 128×75 and 256× 150 grid points and zero order extrapolation (BC2) and second extrapolation(BC3) on a 256×150 grid. Case of a Rossby wave packet with k1 ≡ 1 and k2 ≡ 4.

FIGURES 65

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(A)

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(B)

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(C)

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(D)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Fig. 5. 2d structure of the flow for the Rossby wave packet with k1 ≡ 1 and k2 ≡ 4 at time t = 20 days. Color contourof the vorticity and velocity profile (arrows). exact solution (A), and central incompressible scheme with fixed Dirichlet(B), zero order extrapolation (C), and second order extrapolation (D) boundary conditions at the walls.

66 FIGURES

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(A)

−0.1

−0.05

0

0.05

0.1

0 5 10 15 20 25 30 35−5

0

5

Y (

1000

km

)

(B)

−0.1

−0.05

0

0.05

0.1

Fig. 6. Same as in Figure 5 but with a zonal wavenumber k1 ≡ 4. Exact solution (A) and numerical solution withDirichlet boundary conditions (B).

FIGURES 67

0 2 4 6 8 10 12 14 16 18 200.85

0.9

0.95

1

1.05

1.1

Time (days)

EN

ER

GY

(no

rmal

ized

)

× 79.2104 J/kg Total dry× 49.7016 J/kg Baroclinic× 29.5089 J/kg Barotropic

Fig. 7. Time history of the total (solid), the baroclinic (dashed), and the barotropic (dash-dotted) energies, normalizedby their corresponding initial values, for the barotropic-baroclinic interacting solitary wave.

68 FIGURES

0 10 20 30−5

0

5

Baroclinic

θ

−0.8

−0.6

−0.4

−0.2

0 10 20 30−5

0

5

Barotropic

ωba

r

−0.5

0

0.5

0 10 20 30−5

0

5

−0.5

0

0.5

0 10 20 30−5

0

5

X (1000 km)

Y (

1000

km

)

−0.6

−0.4

−0.2

0

E E

N N

Fig. 8. Baroclinic and barotropic structure of the solitary wave at time t = 2 days. Comparison of the KdV-asymptoticsolution (top panels) and the numerical solution (bottom panels). Contours of the potential temperature and those ofthe vorticity, respectively, on top of the corresponding velocity profile.

FIGURES 69

0 10 20 30−5

0

5

Baroclinic

θ

−0.6

−0.4

−0.2

0

0 10 20 30−5

0

5

Barotropic

ωba

r

−0.5

0

0.5

0 10 20 30−5

0

5

−0.8

−0.6

−0.4

−0.2

0

0 10 20 30−5

0

5

−0.5

0

0.5

0 10 20 30−5

0

5

Y (

1000

km

)

−0.8

−0.6

−0.4

−0.2

0

0.2

0 10 20 30−5

0

5

X (1000 km)

−0.5

0

0.5

5.7

11.5

17

5.7

11.5

17

Fig. 9. Same as Figure 8 but for the numerical solution at times t = 5.7, 11.5, 17 days.

70 FIGURES

Dry region Dry region

Slow moistening frontDrying front

Moist region

s<cm

cm

<s<cd

Fig. 10. Sketch of the dying moisture wave: two interfaces separating the dry regions and the moist (shaded) region.They are moving in the same direction with different speeds (slow moistening and drying fronts). The thick arrowsrepresent the direction and magnitude, s, of the front speeds.

FIGURES 71

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8

9

10

X: longitude (1000 km)

Tim

e (d

ays)

linear extrap. BC, Zonal velocity contour interval: 4 m/s

−14

−18

−22

−26−50

−66

2

−2

−6

−14

−22

Pr (

K/d

ay)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fig. 11. Validation of the dying moisture wave with τc = 2 hours, linear extrapolation boundary conditions at x = 0, X.Contours of zonal velocity, u, color-filled contours of precipitation, and position of the drying and slow-moistening frontsas predicted by the theory from [24] (dash and dash-dot lines)

72 FIGURES

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8

9

10

X: longitude (1000 km)

Tim

e (d

ays)

Periodic BC, Zonal velocity contour interval: 2 m/s

−12

−10−8

−6

−4

−2

−16

−14

−12

−10

−8

−6

−4

−2

0 2

Pr (

K/d

ay)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fig. 12. Same as Figure 11 but with periodic boundary conditions at x = 0, X. An eastward (rarefaction) and awestward (shock) dry gravity waves are created by the jump at x = 0, X caused by the periodic boundary conditions.

FIGURES 73

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

3

4

5

U

Y (

1000

km

)

Fig. 13. Barotropic trade wind.

74 FIGURES

0 2 4 6 8 10 12 14 16 18 20340

360

380

400

420

440

460

480

500

time (days)

EN

ER

GY

(J/

kg)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

Pre

ricip

. (K

/day

)

(A)

0 2 4 6 8 10450

460

470

480

time (days)

EN

ER

GY

(J/

kg)

0 2 4 6 8 100

2.5

5

7.5

10

12.5

15

Pre

ricip

. (K

/day

)

(B)

0 2 4 6 8 10 12 14 16 18 20400

450

500

time (days)

EN

ER

GY

(J/

kg)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Pre

ricip

. (K

/day

)(C)

0 2 4 6 8 10 12 14 16 18 200.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Time (days)

EN

ER

GY

(no

rmal

ized

)

× 27 J/kg Total dry× 21.1556 J/kg Baroclinic× 5.8445 J/kg Barotropic

(D)

Fig. 14. History of total (dry+moist) energy (dashed) and precipitation rate (solid) for the dying moisture waveexperiment: (A) with a trade wind barotropic advection, (B) with beta effect, and (C) with both barotropic advectionand beta effect. The panel (D) display the time histories of the total dry energy (solid) and the baroclinic (dash) andbarotropic (dash-dot) contributions for the case in (C).

FIGURES 75

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0

Time:0 dys,0 hrs,0 mns; CI:0.375 g/kg, filled: 0 0.6704 , 1.3408 , 2.0112 K/day

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

1

Time:1 dys,−1 hrs,47 mns; CI:0.5 g/kg, filled: 0 2.4166 , 4.8332 , 7.2498 K/day

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

3


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

6


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

9


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

13

Time:13 dys,−1 hrs,47 mns; CI:0.375 g/kg, filled: 0 0 , 0 , 0 K/day

Fig. 15. Contours of the water content, q, (g/kg) and precipitation rate, P , (color filled, K/day) for the dying moisturewave experiment with trade wind barotropic advection. Times t = 0, t = 1 day, t = 3 days, t = 6 days, t = 9 days, andt = 13 days.

76 FIGURES

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:1

Time:0 dys,1 hrs,−20 mns; CI:0.5 g/kg, filled: 0 0.21878 , 0.43755 , 0.65633 K/day

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:7


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:14


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:20


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:24


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

1:5


Fig. 16. Same as in Figure 15 but for the experiment with the beta effect at times 1 hour, 7 hours, 14 hours, 20 hours,1 day, 1 day and 5 hours, 1 day and 15 hours, 2 days, 2 days and 6 hours, and 10 days.

FIGURES 77

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

1:15


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

1:24

Time:1 dys,24 hrs,−1 mns; CI:0.625 g/kg, filled: 0 0 , 0 , 0 K/day

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

2:6


0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)Y

(10

00 k

m)

9:24


Figure 16 (continued).

78 FIGURES

0 5 10 15 20 25 30 35−5

−4

−3

−2

−1

0

1

2

3

4

5Time:0 dys,20 hrs,−9 mns; CI:1.174 1/day Max V:6.1001 m/s

X: (1000 km)

Y: (

1000

km

)

0 5 10 15 20 25 30 35−5

−4

−3

−2

−1

0

1

2

3

4

5

X: (1000 km)

Y: (

1000

km

)

Time:0 dys,20 hrs,−9 mns; CI:0.80272 K, Max V:17.7082 m/s

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

0:20


Fig. 17. Profile of the barotropic (top) and the baroclinic (middle) flows on top of the contours of vorticity and potentialtemperature, respectively, and contours of the vertically averaged moisture content, q, on top of , color filled contours ofthe precipitation rate (bottom). Experiment with both beta effect and barotropic advection (trade wind initially). Timet = 20 hours.

FIGURES 79

0 5 10 15 20 25 30 35−5

−4

−3

−2

−1

0

1

2

3

4

5

X: (1000 km)

Y: (

1000

km

)

Time:9 dys,24 hrs,−15 mns; CI:1.175 1/day, Max V:11.13 m/s

0 5 10 15 20 25 30 35−5

−4

−3

−2

−1

0

1

2

3

4

5

X: (1000 km)

Y: (

1000

km

)

Time: 9 dys,24 hrs,−15 mns; CI:0.47998 K, Max V:16.1 m/s

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

X (1000 km)

Y (

1000

km

)

9:24


Fig. 18. Same as Figure 17 but at time t = 10 days.

80 FIGURES

List of Tables

1 L1-norm convergence for the Kelvin , Yanai, and MI=2 Rossby wave with exact-solutionboundary conditions at the zonal walls. CFL=.9 with MC limiter. . . . . . . . . . . . 81

2 Role of boundary conditions and distance from walls for the MI=1 Rossby wave. L1-norm error between exact and computed solution. CFL=.9 with MC limiter are used. 82

3 L1-norm error between the exact and the numerical barotropic velocity fields obtained bythe incompressible central scheme in a zonal channel using the fixed Dirichlet boundaryconditions at the walls. Case of a Rossby wave packet with k1 ≡ 1 and k2 ≡ 4. . . . . 83

TABLES 81

Table 1. L1-norm convergence for the Kelvin , Yanai, and MI=2 Rossby wave with exact-solution boundary conditionsat the zonal walls. CFL=.9 with MC limiter.

Wave integration time 128 × 75 256 × 150 512 × 300

Kelvin (balanced) 2 days 1.2424 E-04 3.0952 E-05 7.7822 E-06

Yanai (unbalanced) 2 days 2.4382 E-04 6.0839 E-05 1.534 E-05MI=2 Rossby (weakly

balanced)47 days 2.1562 E-02 5.3672 E-03 1.3499 E-03

82 TABLES

Table 2. Role of boundary conditions and distance from walls for the MI=1 Rossby wave. L1-norm error between exactand computed solution. CFL=.9 with MC limiter are used.

Boundary condition Y (1000 km) 128 × 75 256 × 150 512 × 300

BC#3 5 4.8899 E-03 1.2198 E-03 3.0610 E-04

BC#1 5 6.1834 E-03 3.8832 E-03 4.5660 E-03

BC#2 5 7.5989 E-03 5.0276 E-03 5.3217 E-03

BC#2 8 7.6370 E-03 1.8871 E-03 4.69759 E-04

TABLES 83

Table 3. L1-norm error between the exact and the numerical barotropic velocity fields obtained by the incompressiblecentral scheme in a zonal channel using the fixed Dirichlet boundary conditions at the walls. Case of a Rossby wavepacket with k1 ≡ 1 and k2 ≡ 4.

Grid 5 days 10 days 15 days 20 days

128×75 0.0087734 0.016355 0.024109 0.031622

256×150 0.0047945 0.0089266 0.013293 0.017545

a high resolution balanced scheme for an idealized

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