Sensitivity of the mesoscale quasi-geostrophic turbulence to the numerical advection scheme: application to an Antarctic Ocean channel dynamics

Contents

Introduction 1

1 Mathematical setup and the QG system 3

1.1 Rotation, Stratification and the Oceanographic Simplifications 3 1.1.1 Equations of motion in a rotating and spherical Earth . 4 1.1.2 The Primitive Equations and the β-plane approximation 9

1.1.3 Linearized equation of state for the Ocean . . . 14

1.1.4 The Boussinesq approximation . . . 16

1.1.5 The buoyancy frequency . . . 18

1.2 Geostrophic and Thermal Wind balance . . . 20

1.2.1 Geostrophic scaling . . . 21

1.2.2 The Rossby number . . . 22

1.2.3 Geotrophic balance . . . 22

1.2.4 Vertical structure of a geostrophic fluid: the Taylor-Proudman effect . . . 24

1.2.5 Thermal wind balance . . . 27

1.3 Vorticity and Potential Vorticity in GFD . . . 28

1.4 Quasi Geostrophy (QG) . . . 33

1.4.1 Quasi Geostrophic Vorticity equation . . . 35

1.4.2 Quasi Geostrophic Potential Vorticity . . . 37

1.4.3 Summary of the QG equations via PV . . . 40

2 Numerical simulations: framework and routine 43 2.1 Simulations framework . . . 44

2.1.2 Computation methodology, discretization and advec-tion schemes . . . 49 2.1.3 Small scale implicit dissipation . . . 62 2.1.4 Numerical diagnostic : how the dissipation is computed. 63 2.1.5 Numerical Forcing . . . 65 2.2 Simulations set up . . . 68

3 Results 70

3.1 Sensitivity of mesoscale turbulence to the advection schemes . 72 3.1.1 Surface vorticity and PV properties . . . 72 3.1.2 Potential enstrophy budget and spectral signature . . . 82 3.2 Spectral behavior of two solutions : UP3 VS UP5 . . . 92 3.3 Timeseries analysis and Statistical convergence . . . 99 3.4 CFL sensitivity . . . 104

Conclusions 111

Introduction

Computational fluid dynamics plays today a central role in understanding geophysical phenomena, such as oceanic dynamics and its turbulent pro-cesses. The interest in studying sea dynamics is growing in several fields, from the natural and climate sciences to the marine-off shore engineering. CFD offers the possibility to overcome some of the limitations of remote sensing systems for ocean detection and in-situ measurements, especially in those areas where the use of these techniques is dangerous or not easily fea-sible, i.e. in the Antarctic Ocean.

In this context, the use of an appropriate numerical model is a crucial issue. Current ocean models, like the Regional Ocean Modeling System (ROMS), provide solutions for fluid dynamics, tracers, sea-ice and biochem-istry equations, and they also include a specific component for numerical advection. We focused our attention on this one.

We compared a set of advection schemes using a 3D Quasi-Geostrophic (QG) model in a forced-dissipated (FDP) turbulent regime with implicit dissipation, applied to the study of a simplified channel of the Antarctic Ocean. The main target was to explore the effects of the numerical schemes and their implicit dissipation on the turbulence field and the resolved scales.

The interactions of mesoscale eddies cause the variance of potential en-strophy to be transferred toward small scale: to prevent an increase of

po-tential enstrophy at the grid size scale, models should dissipate it. This is the point of view from which the numerical models have been evaluated in our study.

We decided to compare some 2nd order Monotone Upstream Schemes for Conservation Laws (MUSCL)[2] with three different flux limiters - mc, super-bee, vanleer -, the 3rd order ROMS-type, two Upwind 3rd and 5th order, and a 5th order Weighted Essentially Non Oscillatory (WENO)[7]_{. These schemes}

have been often tested in Initial Value Problems, finding that high order To-tal Variation Diminishing (TVD) schemes are good to maintain coherent structures[8]. We tested how do they work in a fully developed turbulent regime in FDP, we assessed the behavior with resolution and diagnosed the spectral signature of the implicit dissipation for each scheme. A method to compute implicit dissipation is proposed. Statistical convergence is also assessed, as well as robustness with CFL.

The first chapter illustrates how the general fluid dynamics equations have been specialized to geophysical flows, with our particular application in Oceanic fluid mechanics[4]_{. In this first part, the continuous}

mathemat-ical framework of the study will be also derived and the involved physmathemat-ical processes explained.

The second chapter is dedicated to the numerical part. The domain, the QG numerical model and the computation methodology will be introduced, as well as the advection schemes used for the sensitivity analysis. Numerical forcing and small scale dissipation diagnostic will be analyzed in detail.

The third chapter finally shows and analyzes the results obtained: vor-ticity and potential vorvor-ticity maps, features, spectral signature and potential enstrophy budget, as well as timeseries, statistical convergence, and robust-ness with CFL.

Chapter 1

Mathematical setup and the

QG system

1.1

Rotation, Stratification and the

Oceano-graphic Simplifications

The two main effects that most differentiate oceanic fluid dynamics from other branches of fluid dynamics are the ones caused by rotation and strati-fication. The ocean is a shallow layer of fluid on a sphere: its depth is much less than its horizontal extent, and its motion is strongly affected by Earth rotation and by water vertical gradient of density that is often larger than the horizontal gradient. In the following part we will show how these two effects are modeled in the equations of motion and we will later discuss some possible simplifications.

1.1.1

Equations of motion in a rotating and spherical

Earth

Let us now explain how equations of motion change due to Earth’s rotation. The main effect due to rotation is the occurrence of a Coriolis force and a centrifugal force. The rate of change of a vector in an inertial frame can be linked to the one in the rotating frame as

(dr dt)i = (

dr

dt)r+ Ω × r (1.1)

Where r is the position of a particle and Ω is the angular velocity of the rotating frame. In terms of velocities we have

vi = vr+ Ω × r (1.2)

Where vr is the relative velocity and vi is the inertial one. Expressing

the rate of change of vr in the same manner leads to

(dvr dt )i = ( dvr dt )r+ Ω × vr (1.3) or even (d(vi− Ω × r) dt )i = ( dvr dt )r+ Ω × vr (1.4) (dvi dt )i = ( dvr dt )r+ Ω × vr+ dΩ dt × r + Ω × dr dt (1.5)

Assuming the rate of rotation Ω constant , and exploiting that (dr dt)i = (dr_dt)r+ Ω × r = vr+ Ω × r , we finally obtain (dvr dt )r = ( dvi dt )i− 2Ω × vr− Ω × (Ω × r) (1.6)

The term on the left-hand side is the rate of change of the relative velocity as measured in the rotating frame - the Earth - . The first term on the right-hand side is the rate of change of the inertial velocity, as measured in the inertial frame. The second and third terms are the Coriolis force and the centrifugal force per unit mass. Neither of these are true forces, meaning that when a body is observed from a rotating frame it seems to behave as if these unseen forces affect its motion. More commonly we write the previous as (dvr

dt )r+ 2Ω × vr+ Ω × (Ω × r) = ( dvi

dt )i and we denote the second and third

terms on the left-hand side Coriolis acceleration and centrifugal acceleration respectively.

To write the momentum equation for the mesoscale dynamics on a rotat-ing frame we also make use of some additional hypotesis[4]: the flow is incom-pressible, the viscous terms are negligible so that we can write the equation for the inviscid case, and the mass forces are conservative. Additionally, due to the effect of Coriolis and centrifugal accelerations, the momentum equation in the rotating frame becomes

Dv

Dt + 2Ω × v = − 1

ρ∇p − ∇φ (1.7)

Where the centrifugal term is written as the gradient of a scalar potential and is included in the potential term. We omitted the subscript r since it is now clear that it refers to the rotating frame. Regarding mass and any

other scalar tracer whose measured value is the same in rotating and inertial frame, the conservation equation is not affected by rotation . Indeed , let Φ be a generic scalar field , in the inertial frame it obeys

DΦ

Dt + Φ∇ · vi = 0 (1.8)

For a material derivative we have DΦ_Dt = ∂Φ_∂t + v · ∇Φ, and the individual components of a material derivative are different in the rotating and inertial frame , indeed

(∂Φ ∂t)i = (

∂Φ

∂t)r− (Ω × r) · ∇Φ (1.9)

Where (Ω × r) is the velocity in the inertial frame of a uniformly rotating body. Is also true that

vi· ∇Φ = (vr+ Ω × r) · ∇Φ (1.10)

Adding the last two gives (∂Φ_∂t)i+ vi· ∇Φ = (∂Φ_∂t)r− (Ω × r) · ∇Φ + (vr+

Ω × r) · ∇Φ that leads to

(DΦ Dt)i = (

DΦ

Dt )r (1.11)

The material derivative of the scalar field is frame invariant. Furthermore, ∇ · vi = ∇ · (vr+ Ω × r) = ∇ · vr , because ∇(·Ω × r) = 0 . This, together

with (1.11) confirms that rotation does not influences the mass coservation equation.

The other factor that has some consequences on the equations of mo-tion is the nearly-spherical shape of the Earth. The topic is quite delicate because the presence of a centrifugal force causes some complications when using spherical coordinates. For our purposes it is not necessary to enter into heavy details, that can be found in litterature [4]_{. We can just mention}

that the centrifugal force is a potential force, like gravity, thus is a common practice to define an effective gravity which is the sum of the true, or Newto-nian gravity and centrifugal force. NewtoNewto-nian gravity is directed toward the center of the Earth with small deviations due to the pole-oblateness. The line of action of the effective gravity differs in general slightly from this, and therefore there is a component of effective gravity in the horizontal plane(the plane perpendicular to radius direction). Both true gravity and centrifugal force are potential forces, thus we may define a geopotential as the potential of the effective gravity, and observe that iso-geopotential surfaces are not spherical because the centrifugal force varies with latitude. The problem is that, using spherical coordinates, the components of the momentum equation would be dominated by the presence of the horizontal component of effective gravity, thus by a static balance between a pressure gradient and gravity (ra-dial direction) or centrifugal force (horizontal). In principle there is nothing wrong with this but it obscures the horizontal balance involving Coriolis force ( much smaller ) and pressure gradient, that is instead what really counts in terms of large scale horizontal flows. We can simply overcome this problem by choosing the direction of effective gravity to define the vertical direction , meaning that the horizontal component of it is identically zero. This trick trades a potentially large physical error for a very small geometric error.

We now introduce the general form of the equations of motion in spherical coordinates. Since our purpose is to derive a simplified model we will not dwell on the mathematical passages that leads to them, also because it is a

well treated topics easily accessible by literature [4]_{. Let us just write them}

and go further with the important simplifications.

Mass Conservation Equation (MCE) : assuming that λ is longitude, θ is latitude, r is the Earth radius modulus, one possible form of MCE is

∂ρ

∂t + ∇ · (ρv) = 0 , that becomes the following :

∂ρ ∂t + 1 cosθ ∂(uρ) ∂λ + 1 cosθ ∂ ∂θ(vρcosθ) + 1 r2 ∂ ∂r(r 2_{wρ) = 0 (1.12)}

Tracer advection equation ( i.e thermodynamics equation) where T is the generic tracer, follows :

DT Dt = ∂T ∂t + u rcosθ ∂T ∂λ + v r ∂T ∂θ + w ∂T ∂r = 0 (1.13)

The Momentum Equation (ME), that has the vectorial form (1.7), can be written in scalar components as follows, after some manipulation:

Du

Dt − (2Ω + u

rcosθ)(vsinθ − wcosθ) = − 1 ρrcosθ ∂p ∂λ (1.14) Dv Dt + wv r + (2Ω + u rcosθ)(usinθ) = − 1 ρr ∂p ∂θ (1.15) Dw Dt − u2_{+ v}2 r − 2Ωucosθ = − 1 ρ ∂p ∂r − g (1.16)

Where the terms involving Ω are the Coriolis terms. Note that the viscous form is given just adding a frictional term Fx on the righthand side of (1.14),

1.1.2

The Primitive Equations and the β-plane

approx-imation

At this point we are able to introduce the so-called primitive equations (PE) as a simplification of the previous equations of motion. Three related approximations are involved to obtain PE equations. They all rely on the presumed small aspect ratio of the oceanic motion, and in fact they have revealed to be very accurate approximation for the large scale ocean dynamics

[4]

- Shallow fluid approximation : The ocean is a shallow layer of fluid in comparison with the Earth radius. With the shallow layer approximation we write the r-coordinate as r = R + z where R is the Earth radius and is a constant and z is a coordinate that increases in radial direction. The coordinate r is thus replaced by R except where it is used as the differentiating argument. For example:

1 r2 ∂(r2_w) ∂r 7−→ ∂w ∂z (1.17)

The ”vertical direction” is z-direction.

- The Traditional approximation : the small terms uw_r and vw_r , and the Coriolis terms in the horizontal momentum equations involving the vertical velocity w are all neglected. In other words the approximation on the Coriolis terms is like we neglect the components of Ω not in the local vertical direction.

- Hydrostatic approximation : The left-hand side terms of equation (1.16), namely the advection of vertical velocity, the u2+v_r 2 term and the Coriolis term, are all neglected. That is to say that the gravitational term is assumed to be balanced by the pressure gradient in vertical direction. Hydrostatic approximation assumes the following form:

∂p

∂z = −ρg (1.18)

This last assumption does need a bit more attention since it leads the vertical momentum equation to assume a very important form, known as Hydrostatic balance. The general form for the momentum equation in the vertical direction is the well known Dw_Dt = −1_ρ∂p_∂z − ρg and considering the relative order of magnitudes of each term we can write it as

W T + U W L + W2 H + ΩU ≈     1 ρ ∂ρ ∂z     + g (1.19)

Typical reference values representative of large-scale eddying motion in the Ocean are:

Horizontal length scale L ≈ 105 _m

Horizontal velocity U ≈ 0.1 m s−1

Time scale T ≈ 10 days (106 _s)

Vertical length scale H ≈ 103 _m

Vertical velocity W ≈ 10−4 m s−1

Using this typical values for the scaling (1.19) shows that the pressure term is the only one which can balance the gravitational one. A general condition for hydrostasy to hold is that the aspect ratio (H_L)2 _{<< 1,}

cer-tainly satisfied in ocean mesoscale dynamics. Thus, in this sense even the hydrostatic balance is a small aspect ratio approximation.

Making the above approximations (hydrostatic, shallow fluid and tradi-tional) leads to a slightly simplified form of the equations of motion. The

momentum inviscid equations (1.14)-(1.16) become Du Dt − 2Ωsinθv − uv R tanθ = − 1 Rρcosθ ∂p ∂λ (1.20) Dv Dt + 2Ωsinθu + u2 Rtanθ = − 1 Rρ ∂p ∂θ (1.21) 0 = −1 ρ ∂p ∂z − g (1.22)

Note that the last one is right the hydrostatic balance. A very important parameter can now be defined , thanks to the presence of the term 2Ωsinθ in both equations (1.19) and (1.20) : this term is the so-called Coriolis pa-rameter f , defined indeed as

f = 2Ωsinθ (1.23)

The mass conservation equation after the above approximations and some manipulations becomes: ∂ρ ∂t + 1 Rcosθ ∂uρ ∂λ + 1 Rcosθ ∂(vρcosθ) ∂θ + ∂wρ ∂z = 0 (1.24)

That in compact form is

Dρ

Dt + ρ∇ · v = 0 or

∂ρ

∂t + ∇ · (ρv) = 0 (1.25)

A compact vector form for the momentum-PE can be written involving one vector equation for the horizontal, while the vertical is still the hydro-static equation (1.22). Thus for the horizontal is

Du

Dt + f × u = − 1

ρ∇zp (1.26)

where u = ui + vj + 0k is the horizontal velocity, f = f k = 2Ωsinθk and ∇zp = [_Rcosθ1 ∂p_∂λ;_R1∂p_∂θ] is the gradient at constant z.1em At this point we

have a set of simplified equations that retain both sphericity and rotational effects; although certain approximations have been made the model still has got a quite heavy mathematical structure so that is reasonable wondering if the above effects are both important to describe the dynamics or not.

Indeed, while the rotation of the Earth plays a key role in many dynamical processes, the sphericity is not always so, especially for phoenomena occur-ring at scales somehow smaller than global, for example at the mesoscale.

This allows us to use a locally Cartesian representation of the equations, that leads to another important simplification: the beta plane approxima-tion. Let us first define the tangent plane as a plane tangent to the Earth surface at a certain latitude θ0, and use Cartesian coorinates (x, y, z) to

de-scribe the motion on that plane. For ”small” displacements on the tangent plane (i.e. the mesoscale can be consider such small compared with the global scale) the cartesian coordinates do not differs significantly from the spherical ones, that is to say

(x, y, z) ≈ (Rλcosθ0, R(θ − θ0), z) (1.27)

velocity in the tangent plane, in approximately est-west direction the former and north-south the latter , while w is the vertical component of velocity in the tangent plane. Components u , v and w are even called respectively zonal, meridional and vertical velocity - they are scalar components -. Recalling the definition of Coriolis parameter (1.23) is clear that on the tangent plane f = 2Ωsinθ0 = f0 is a constant since this plane is defined at a certain latitude

θ0. To improve this feature we can note that for not big variation in latitude

f = 2Ωsinθ ≈ 2Ωsinθ0+ 2Ω(θ − θ0)cosθ0 (1.28)

Then it is possible to mimic on the tangent plane this behavior with a linear variation of f with y. That is to say

f = f0+ βy (1.29)

Where β = ∂f_∂y. This relation represent the important approximation known as the beta-plane approximation : this capture the most important dynamical effects of sphericity without the complicating geometric effets, which are not essential to dynamical phoenomena.

Including the beta-plane approximation along with Hydrostatic, Shallow fluid and Traditional approximations, the momentum inviscid equations are now Du Dt − f v = − 1 ρ ∂p ∂x (1.30) Dv Dt + f u = − 1 ρ ∂p ∂y (1.31)

0 = −1 ρ

∂p

∂z − g (1.32)

In compact vector form, considering also viscosity and body forces, they become: Dv Dt + f × v = − 1 ρ∇p + ν∇ 2_{v + F} b (1.33)

Where ν ≡ µ_ρ is the kinematic viscosity and Fb is any body force per unit

mass, such as g. Mass conservation and thermodynamic equations are just the PE form of them, specialized in the cartesian frame of reference on the tangent plane. To complete the set of equations for the ocean dynamics the last step is to express an equation of state for sea water, as shown in next section 1.1.3 .

1.1.3

Linearized equation of state for the Ocean

The equation of state for sea water is a relation ρ = ρ(T, S, p), where T is the water temperature, S is the Salinity and p is the pressure. The salinity S is a measure used to parametrize the total concentration of several types of salt in the sea water. Indeed, water density in the ocean is significantly affected by dissolved salt: sea water is a solution of many ions in water - chloride(≈ 1.9% by weight) , sodium(1%), sulfate(0.26%), magnesium(0.13%), and others -with a total average concentration of about 35 ppt. The ratio of the fractions of these salt are more or less constant throughout the ocean, and their total concentration is parametrized by S.

The equation of state is known only by empirical evaluations, usually in the form of a polynomial expansion series in powers of the departures of the

state variables from a specified reference state. For the purposes of physical oceanography it is reasonable to use a simplified linearized form:

ρ = ρ0[1 − α(T − T0) + β(S − S0)] (1.34) Where α = −1 ρ ∂ρ ∂T (1.35)

is the ”Thermal expansion coefficient” for sea water, and

β = 1 ρ

∂ρ

∂S (1.36)

is called ”haline contraction coefficient for sea water”.

The linearization is made for fluctuations around a reference state of (ρ0, T0, S0), and the partial derivatives are computed while the other state

variables held constant. Typical oceanic values for this reference are resumed in table (1.1). One of the typical features of marine currents is that their horizontal dynamics is much more significant than their vertical motion. Fur-thermore, if any significant vertical motion is observed it is limited both in time and spatial extension. A strong pressure gradient is quite rare, so it is reasonable to neglect its influence on density in the equation of state, at least for our scopes. The linearized equation of state can be used in situation when either the spatial extent of the domain is not so large as to involve significant changes in the expansion coefficients or when the qualitative behavior of the flow is not controlled by the quantitative details of the equation of state.

When this linearization can not be applied (i.e. cabelling instability, thermobaric instability, etc.), various forms of the equation of state have been proposed, in general including non linear effects into secondary expansion coefficients.

Parameter Description Value

ρ0 Reference Density 1027 kg m−3

T0 Reference Temperature 283 K

S0 Reference Salinity 35 ppt

α thermal expansion coefficient 2 10−4K−1 β haline contraction coefficient 8 10−4ppt−1

Table 1.1: Typical values of expansion-contraction coefficients, and equation-of-state parameters for sea water at the reference equation-of-state.

1.1.4

The Boussinesq approximation

The density variations in the ocean are quite small compared to the mean density, which represents a reference state. Exploiting this feature of ocean density we can derive a simple but still accurate form to describe the density profile. Let us first examine how much density does vary in the Ocean.

Density variations in the ocean are due to three effects: the thermal expansion of water if its temperature changes (which we will denote ∆Tρ), the

haline contraction if salinity changes (∆sρ), and if we consider a very accurate

equation of state we include also the compression of water by pressure (∆pρ).

An appropriate equation of state to evaluate these effects has been derived in 1.1.3. If we include also the pressure term, it is

ρ = ρ0  1 − α(T − T0) + β(S − S0 + p ρ0c2s )  (1.37)

Where cs = 1490 ms−1 is the reference speed sound. We can evaluate

Pressure compressibility: we have ∆pρ ≈ ∆p/c2s ≈ ρ0gH/c2s , where H

is the depth and we used hydrostatic approximation to evaluate pressure changes. |∆pρ ρ0 | << 1 if gH c2 s << 1 (1.38)

Or even H << c2_s/g. The quantity c2_s/g ≈ 200km is the density scale height of the ocean. Thus the pressure at the bottom of the ocean, say at H = 10km in the deep trenches, is insufficient to compress the the water enough to make significant changes in its density. Changes in density due to dynamical variations of pressure are small if the Mach numbes is small, and this is also the case.

Thermal expansion: We have ∆Tρ0 ≈ αρ0∆T , and therefore

|∆Tρ ρ0

| << 1 if α∆T << 1 (1.39)

For δT = 20 K, is α∆T ≈ 4 × 10−3, and evidently we would require temperature differences of order 5000 K to obtain order one variations in density.

Saline contraction: We have ∆sρ0 ≈ βρ0∆S, and so it is

|∆sρ ρ0

| << 1 if β∆S << 1 (1.40)

5 × 10−3 , the fractional change in density of seawater is correspondingly very small.

Evidently, fractional density changes in the ocean are very small, and indeed exploiting the smallness of these variations we can apply Boussinesq approximation: it is possible to write the equations of motion in a simplified form considering density as a constant, with a still acceptable accuracy (error of 0,02 - 0,03). A useful way to see Boussinesq approximation is to set

ρ = ρ0+ δρ(x, y, x, t) ≈ ρ0 because δρ(x, y, x, t) << ρ0 (1.41)

Where ρ0 is the reference constant state and δρ(x, y, x, t) represents

den-sity variations in term of a small perturbation around the basic state. Notice that, associated with the reference density state, there is also a reference pressure that is defined to be in hydrostatic balance with it.

1.1.5

The buoyancy frequency

In this section we will define two parameters that will appear later in the quasi geostrophic model: the Brunt-Vasala frequency N , and the buoyancy b. Our focus is on how a fluid might oscillate if it were perturbed away from a resting state: we consider vertical displacements, and the restoring force is gravity. We will neglect the effects of rotation because they are not crucial to understand the main concept. Initially we will neglect all the horizontal motion entirely: we consider a fluid initially at rest in a constant gravitational field, thus in hydrostatic balance. Suppose that a small parcel is adiabatically displaced upward by the small distance δz, without altering the overall pressure field, so the parcel instantly assumes the pressure of its environment.

If, after the displacement, the parcel is lighter than its environment then it will accelerate upwards because the upward pressure gradient force is now greater than the downward gravity force on the parcel. The parcel is buoyant and the fluid is statically unstable.

If on the other hand the parcel find itself heavier than the surrounding environment, then the gravitational force on the parcel is greater than the upward pressure gradient force, thus the parcel move downward and the fluid sink back toward its initial position: an oscillatory motion will develop. Such an equilibrium is statically stable. Using this simple concept, we will define the parameters at our interest. Due to the quasi-incompressibility of sea water, we will assume it as incompressible (Boussinesq assumption). Let us consider the basic case of the parcel displacement when its density is conserved, that is Dρ_Dt = 0.

The environmental density profile is ρ∗(z), and the parcel density is ρ, and let z be the initial level. Thus, the initial density of the parcel is ρ(z) = ρ∗(z), same as the environment. A displacement toward a new level z + δz occurs. The parcel density is still ρ∗(z), but the density of the surrounding environment change of an amount δρ expressed as follows:

δρ = ρ(z + δz) − ρ∗(z + δz) = ρ∗(z) − ρ∗(z + δz) = −∂ρ

∗

∂z δz (1.42)

If ∂ρ_∂z∗ < 0 the parcel will be heavier than the surroundings and therefore the parcel displacement will be stable. If, on the other hand, ∂ρ_∂z∗ > 0 then it will be unstable. The upward force per unit volume on the parcel is given by F = −gδρ = g∂ρ_∂z∗δz. The motion of the parcel is described by Newton’s second law : ρ(z)∂_∂t2δz2 = g ∂ρ∗ ∂z δz . Or even: ∂2_δz ∂t2 = g ρ∗ ∂ρ∗ ∂z δz = N 2_{δz (1.43)}

Where finally we defined the buoyancy frequency N so that

N2 = −g ρ∗

∂ρ∗

∂z (1.44)

For this problem, if N2 _{> 0 the density profile is stable and N is the}

frequency of stable oscillations. In other words, the condition for stability is that density decreases moving vertically toward the surface, and it is a function of z alone. Finally the buoyancy b is defined as follows

b = −gδρ ρ0

(1.45)

Where ρ0 is a reference density, that for ocean water can be assumed as

in table 1.1

1.2

Geostrophic and Thermal Wind balance

We now consider the dominant dynamical balance in the horizontal compo-nents of the momentum equation. In the horizontal plane the Coriolis term is much larger than the advective terms, and the dominant balance is between it and the horizontal pressure term. This balance is called geostrophic balance, and it occurs under certain dimensional features, as we now investigate.

1.2.1

Geostrophic scaling

Let us now analyze the order of magnitudes of all the terms involved in the horizontal component of the momentum equations, i.e. the first two scalar equations of (1.33). We will focus only on the first one since the procedure is exactly the same for both. The target is to investigate if any of the terms is negligible in respect to some other involved scale.

We re-write the first of (1.33) in term of order of magnitudes as

U ∆t+ U2 L + U2 L + U W H = f U + 1 ˜ ρ ˜ p ˜ x + ν U L2 + ν U L2 + ν U H2 (1.46)

The first term is evaluable by _∆tU ≡ U2

L , The fourth term is U W

H ≡

U2

L,

the first two viscous terms are at most ν_LU2 ≡

U2

L because ν ≈< U L. The

third viscous term is again ν_HU2 ≡

U2

L because also for the vertical mixing

of the horizontal velocity is ν ≈< W H, leading to the the same size of the other viscous terms. Note that the size of the pressure term is not explicitly described because it is assumed to balance any other acceleration resulting from all the remaining terms. This is actually what happens in a geostrophic flow. Finally, we are able to evaluate the relative importance of the scales by using typical values expressed in tab (1.2), and we find that the Coriolis term dominates on the other acceleration term, that is to say that rotation effects are important and both viscous and inertial terms in the momentum equation are negligible. The only term able to balance this dominant Coriolis acceleration is indeed the pressure term.

The geostrophic scaling just presented is the core of the following concepts that will lead to our final mathematical set.

Variable Scaling Symbol Meaning Value

(x,y) L Horizontal length scale 105_m

t T Time scale 106s

(u,v) U Horizontal velocity 0.1 ms−1

Ro Rossby Number 0.01

f Coriolis parameter 10−4s−1

Table 1.2: Typical magnitudes of large scale flows in the ocean.

1.2.2

The Rossby number

Let us first introduce an important non dimensional number, that is essen-tially the ratio of the relative acceleration to the coriolis acceleration. This parameter is called Rossby Number and it is defined as

Ro ≡ U

f L (1.47)

The Rossby number characterizes the importance of rotation of a fluid, indeed it arises from a simple scaling of the horizontal momentum equations (1.30) or (1.31) : the magnitude of the coriolis term is f U , while the advective term size is U2_{/L, so that their ratio lead to (1.47).}

When Ro is small then rotation effects are important, and inertial terms in the momentum equations are negligible in the horizontal plane.

1.2.3

Geotrophic balance

The core concept of the geostrophic balance relies on the small size of Rossby Number, much smaller than 1. In tab. 1.2 typical magnitudes of large scale

oceanic flows are given, as well as the consequent Ro size. A sufficently small Ro makes the rotation term to dominate the nonlinear advection term, and if the time scale of the motion scales advectively then the rotation term also dominates the local time derivative. If we are neglecting the smaller terms, thus we consider the so-called zero-order contributes in the balances, i.e. the geostrophic components, the only term that can balance the rotation term is the pressure one, and therefore we have

f × u ≈ −1 ρ∇p (1.48) or , in scalar components f u ≈ −1 ρ ∂p ∂y f v ≈ 1 ρ ∂p ∂x (1.49)

This important balance is the so-called Geostrophic Balance and its pro-found consequences give to geophysical fluid dynamics some unique charac-teristics in the broad field of fluid dynamics. At this point is clear that we can define geostrophic velocity the horizontal velocity field

f ug ≡ − 1 ρ ∂p ∂y f vg ≡ 1 ρ ∂p ∂x (1.50)

and for low Ro is of course u ≈ ug and v ≈ vg, where ug and vg are the

geostrophic velocity scalar components, in horizontal and meridional direc-tion respectively. An important property of the geostrophic field is that, if the Coriolis force is constant and the density does not vary in the horizontal, then the flow is horizontally non-divergent, thus

∂ug

∂x + ∂vg

∂y = 0 (1.51)

We can also define a geostrophic stream function Ψ(x, y, z, t) = p(x,y,z,t)_ρ

0f , so : ug = − ∂Ψ ∂y vg = ∂Ψ ∂x (1.52)

At a fixed depth the derivatives of Ψ give the corresponding horizontal velocity field.

The second remarkable property of a geostrophic velocity field is that it is parallel to isobars, as qualitatively shown in figure (1.1). At a fixed depth the non-equatorial motion of water masses in the Ocean can be described by geostrophic equations and represented by figure (1.1): where isobars are closed the flux will show vortical structures, while if they are open and almost-parallel the flux will behave as a jet. Circulation around low pressure zones is called cyclonic, while the one exsisting around high pressure areas is called anticyclonic. At negative latitudes is f < 0: cyclonic circulation is clock-wise and anticyclonic circulation is anticlockclock-wise. The opposite happens at positive latitudes, because f > 0.

1.2.4

Vertical structure of a geostrophic fluid: the

Taylor-Proudman effect

Figure 1.1: This is a qualitative illustration of geostrophic pressure field and the parallel velocity field. Isolines are isobars (higher pressure in red, lower pressure in blue). Arrows represent velocity vectors. Note that the closer the isobars the higher the pressure gradient, thus the higher the velocity intensity.

v = 1 f0 ∂φ ∂x u = − 1 f0 ∂φ ∂y ∂φ ∂z = −g (1.53)

Differentiating the first two of (1.53) with respect to z, and using the third one of (1.53), yelds

∂v ∂z = − 1 f0 ∂g ∂x = 0 and ∂u ∂z = 1 f0 ∂g ∂y = 0 (1.54)

The geostrophic velocities are non divergent (∇ · ug = 0), thus using

the mass continuity then gives ∂w/∂z = 0. If also the Coriolis term is constant, as it is at this stage of approximation, then the vertical velocity is not a function of depth and none of the components of velocity vary with depth. if there is any solid boundary anywhere in the fluid, for example at the bottom, then here is wg = 0 and therefore wg = 0 everywhere. If it

happens, the motion is actually two dimensional, as it occurs in planes that lie perpendicular to the axis of rotation. This result is known as the Taylor -Proudman effect, namely that for constant density flow in geostrophic and hydrostatic balance the vertical derivatives of the horizontal velocities are zero as well as the vertical velocity.

In real oceanic flows we do not observe exactly such vertical coherent structure, mainly because of the effects of stratification that implies varia-tions in density. However, if the flow is rapidly rotating - that is just another way to refer to small Rossby numbers- then we expect that the horizon-tal flow is in near geostrophic balance , or quasi geostrophic balance, and therefore near divergence free, at least at a zero order approximation; thus ∇ · u << U

L and W << HU

L . Indeed, the existance of ageostrofic components

is a higher order effect,that does not affect the zero-order field. We will treat the quasi-geostrophy in a dedicated section.

1.2.5

Thermal wind balance

Thermal wind balance arises by combining geostrophic balance and hydro-static approximation. We can re-write the geostrophic balance as

f vg =

∂φ

∂x f ug = −

∂φ

∂y (1.55)

And hydrostatic balance being

b = ∂φ

∂z (1.56)

Then we have the thermal wind balance that connects the vertical deriva-tives of geostrophic velocity with horizontal variations of buoyancy

f∂vg ∂z = ∂b ∂x f ∂ug ∂z = − ∂b ∂y (1.57) Or equivalently ∂ρ ∂y = ρ0f g ∂u ∂z ∂ρ ∂x = − ρ0f g ∂v ∂z (1.58)

Obviously this balance is meaningful in real oceanic flows where the ver-tical derivatives of the geostrophic field are not precisely zero and the fluid is stratified. The information hidden into these equations is that, if the isopyc-nals are not parallel to the horizontal then the vertical velocity will vary. The situation expressed by thermal wind balance is typical of non-inertial frames. This balance cannot be realized in inertial frames because the steadiness (in

these frames) demands minimum potential energy, which is achieved only if isopycnals are orthogonal to the local gravity vector.

1.3

Vorticity and Potential Vorticity in GFD

Vorticity and Potential Vorticity both play a central role in geophysical fluid dynamics (GFD). The large scale circulation of the Ocean is in large part governed by the evolution of Potential Vorticity (PV). In this section we will explore in particular the role of PV, giving a careful attention to its conservation law, since it is the base on which the numerical QG code has been built up. To define PV and discuss its conservation property,we will show how it is tied to one of the most important theorem of all of fluid mechanics : Kelvin’s circulation theorem.

First of all we briefly recall some basic definition of fluid dynamics. Vor-ticity is defined to be the curl of velocity

ω ≡ ∇ · v (1.59)

Circulation, C, is defined to be the integral of velocity around a close fluid loop, and for Stoke’s theorem it is linked to vorticity as follows

C ≡ I vdr = Z S ω · dS (1.60)

Where dr is the line element and the the line integration being over a closed material circuit, while S is of course any surface bounded by the path

of the line integral. We are not interested here in demonstrating these two statements, as they are a well covered topic in literature[4].

According with definition (1.59), the vorticity equation is obtained by taking the curl of the momentum equation, using vector identities and ma-nipulations along with the mass conservation, it finally gives

Dω_b Dt = (ω · ∇)v +b 1 ρ2(∇ρ × ∇p) + 1 ρ∇ × F (1.61) Where ω ≡_b ω

ρ and F resumes viscous and body forces. For simplicity we

set now F = 0 and we will assume it in the present section.

We may further simplify this equation by writing it for two dimensional flows, where both tilting and stretching terms vanish, thus (ω · ∇)v van-_b ishes, and where vorticity is perpendicular to velocity. The two dimensional vorticity equation for incompressible flows then becomes

Dω

Dt = 0 (1.62)

Or even, in expanded form

∂ω

∂t + u · ∇ω = 0 (1.63)

That expresses the material conservation of vorticity. Note that, without hurting the validity of this conservation property, in a rotating frame of reference it is useful to refer to the vorticity as the absolute vorticity, due to the sum of a relative vorticity ζ and a planetary vorticity 2Ω, which indeed represents a Coriolis effect. To demonstrate this we refer to literature[4] _.

At this point it is necessary to recall the Kelvin’s circulation theorem: showing why it is not satisfied for atmospheric and oceanic flows we will derive a conservation law that overcomes such limitations. In this context, Potential Vorticity is indeed the protagonist of this new conservation equation.

The most general form of Kelvin’s theorem states that the circulation C around a material loop is invariant for a barotropic fluid that is subject only to conservative forces. The prove is given in literature[4] _{, we just write the}

main passages and the result:

DC Dt = D Dt I v · dr = I −1 ρ∇p · dr (1.64)

Where, using Stoke’s theorem with S any surface bounded by the loop, it is I 1 ρ∇p · dr = Z S ∇ × (∇p ρ ) · dS = Z S −∇ρ × ∇p ρ2 · dS (1.65)

Where the right-most expression above is the integral of the solenoidal vector. The integral evidently vanishes identically if p is a funcion of ρ alone, in which case ∇p is parallel to ∇ρ so their × is identically zero. In this case it is

D Dt

I

v · dr = 0 (1.66)

In baroclinic flows, such as in Ocean and Atmosphere, the circulation is not generally conserved because of the existence of a non-zero solenoidal term

∇ρ × ∇p, and the rate of change of C depends on this term. For baroclinic flows the material derivative of C gives

DC Dt = −

I _∇p

ρ · dr (1.67)

It turns out that it is possible to derive a conservation law that overcomes this failing, that is very useful for geophysical fluid dynamics. This is the conservation of Potential Vorticity, introduced by Rossby and then in a more general form by Ertel5_{. The idea is that we can use a scalar field that is}

being advected by the flow to keep track of the evolution of fluid elements. In baroclinic fluids this scalar field must be chosen to be a function of density and pressure alone. Then using the scalar evolution equation in conjuction with the vorticity equation gives us a scalar conservation equation. There are two main ways to define potential vorticity: the more general Ertel’s definition is

q = ω

ρ · ∇θ (1.68)

where ω is the scalar component of ω = ω · k, the total or absolute vorticity expressed in two dimensional flows by ω = (ζ + f ), and θ is a ma-terially conserved scalar field, for example potential temperature. According to Rossby’s definition the potential vorticity is instead expressed as follows

q = (f + ζ)

h (1.69)

depth of the fluid column. The PV conservation can be derived in some superficially different ways. The demonstration of PV conservation theorem is given in literature[1]; we write here just the result that is what actually interests us.

The conservation law for Potential Vorticity q reads, for conservatives flows Dq Dt = 0 (1.70) Or, expanded ∂q ∂t + u · ∇q = 0 (1.71)

Where q can be expressed from both (1.68) and (1.69). The advection operator u · ∇q, can be rearranged to the convenient Jacobian following form

∂q ∂t + u · ∇q = u ∂q ∂x + v ∂q ∂y = − ∂ψ ∂y ∂q ∂x + ∂ψ ∂x ∂q ∂y = J [φ, q] (1.72)

So that PV conservation is also written in Jacobian form as

∂q

∂t + J [φ, q] = 0 (1.73)

viscous term F in equation (1.61), will lead to a non zero right hand side in (1.70-73).

1.4

Quasi Geostrophy (QG)

The geostrophic and hydrostatic balances presented in the previous section have been derived through an evaluation of magnitude orders in the momen-tum equations. Although they enormously simplify the Primitive Equations, they lead to a non-closed set of equations so it become necessary to consider higher order terms of the governing equations. The full procedure is given in

[3]_{, we resume here the main concepts and results.}

The basic idea is to perform a Taylor expansion of all the fields: assuming that all the small parameter are of the same order  << 1 , all the variables are now written as a series in powers of the small  , starting with a zero-order contribution indicated by subscript 0, and so on. For example velocity components read

u = u0+ u1+ ... , v = v0+ v1+ ... , w = w1+ ... (1.74)

Note that the zero-order contribution of w vanishes because it has a lead-ing term of order Ro, which is a  contribution. Indeed, we will see below that the zero-order horizontal velocity field is divergence free, and furthermore the zero-order feld is in fact the geostrophic one. Inserting all the power expan-sions into the primitive equation system, it is possible to collect the terms of same order in  and require the equations to be satisfied for each order. Fortunately, the two lowest powers are useful for our purposes.To facilitate the notation, we will write the equations of all this section in terms of scaled pressure and density, i.e. p = p/ρ_b 0 and ρ =ρ/ρb 0. The constant factor ρ0 is eliminated. The zero-order momentum equations are

−f0v0 = − ∂p0 ∂x (1.75) f0u0 = − ∂p0 ∂y (1.76) 0 = −∂p0 ∂z − gρ0 (1.77)

and they express the geostrophic and hydrostatic balance of the basic state. Here is u0 = ug and v0 = vg and the geostrophic stream function

Ψ = p/f0 is still defined by (1.52). The horizontal geostrophic velocity is

divergence free

∂u0

∂x + ∂v0

∂y = 0 (1.78)

which agrees with 0-order continuity equation, i.e. the continuity equation does not constrain the lowest order at all. Furthermore, the density balance

Dρ Dt −

N2

g w = 0 does not have a zero order contribution, since w0 vanishes

identically. Thus, the 0-order field variables (u0, v0, ρ0, p0) are more than

the number of equation available to determine them (1.75-77). The problem is incomplete and it is necessary to explore higher orders of the governing equations.

The general form for the 1-order equation of momentum and density balance is, after some manipulations

Du0 Dt − f (v0+ v1) = − ∂p ∂x + Fu (1.79) Dv0 Dt + f (u0+ u1) = − ∂p ∂y + Fv (1.80) 0 = −∂p1 ∂z − gρ1 (1.81)

Dρ0

Dt − N2

g w1 = 0 (1.82)

Where the material derivative operator D/Dt is based solely on the 0-order geostrophic velocities, so there is no vertical advection. We include here also the non conservative terms Fu , Fv. For a detailed description see [2].

The closure comes up this way: the 1-order density equation does not predict the 1-order density, instead it gives the 1-order vertical velocity w1 in terms

of 0-order field (ρ0, u0, v0). The 1-order continuity equation can be used to

evaluate the divergence of the 1-order horizontal velocities; finally, forming the vorticity of the 1-order momentum balances, the 1-order pressure field can be eliminated, and the only remaining 1-order contribution comes from the divergence of the ageostrophic velocities. The ageostrophic or so-called quasi geostrophic vorticity balance, thus yelds a condition on the flow which involves 0-order fields, as we will see below.

1.4.1

Quasi Geostrophic Vorticity equation

The derivation of the ageostrophic vorticity equation from the 1-order equa-tion is straightforward but somewhat lengthy and we drops the mathemathics here, for further details refer to [3]_{. The result is}

D(ζ + f ) Dt = f0

∂w1

∂z + eF (1.83)

Where the term eF stems from the curl of an opportune non conservative term. ζ is the vertical vorticity of the geostrophic flow:

ζ = ∂v0 ∂x − ∂u0 ∂y = ∇ 2 hΨ (1.84)

The ageostrophic component w1 in (1.83) is determined by the 0-order

density (1.82) , that yelds:

w1 = g N2( Dρ0 Dt ) = g N2  D Dt(− 1 g ∂p0 ∂z )  (1.85)

Where we used (1.77) for the last equivalence. Equation (1.83) thus become D(ζ + f ) Dt + f0 ∂ ∂z( D Dt 1 N2 ∂p0 ∂z ) = eF (1.86)

The temporal and material derivatives can commute, and finally using the expression of vorticity via streamfunction (1.84), and recalling that p0 = f0Ψ,

we have finally D Dt  ∇2 hΨ + βy + ∂ ∂z f2 0 N2 ∂Ψ ∂z  = eF (1.87)

The important concept shown by this equation is that in a QG regime the absolute vorticity ζ +f is not materially conserved even with eF = 0 , because of the presence of the stretching term. Instead, the material conservation still holds, in absence of eF , for the whole quantity into brackets. This important quantity is indeed the quasi geostrophic potential vorticity, as we will treat in next section (1.4.2) .

It is remarkable that the stream function Ψ is the only prognostic variable which determines the entire flow at any time, as shown in the relations below (0-index omitted for simplicity):

u = −∂Ψ ∂y v = ∂Ψ ∂x w = − f0 N2 ∂Ψ ∂z (1.88) ρ = −f0 g D Dt ∂Ψ ∂z (1.89)

Just w is 1-order. In fact, the vertical velocity is the only ageostrophic field which is uniquely determined by quasi geostrophic theory; in particular the ageostrophic components of the horizontal velocities and the ageostrophic density and pressure fields remain unknown at this level of approximation.

1.4.2

Quasi Geostrophic Potential Vorticity

As it is now clear, in a real like description of oceanic flows a pure geostrophic assumption is not good enough to describe most of the processes, thus also the ageostrophic first order component must be taken into account. The resulting system describes a Quasi Geostrophic regime, since the ageostrophic part is much smaller than the geostrophic one, that is to say the flow is almost geostrophic but still 3D. In this context it is also clear that the definition of PV given for pure geostrophic flows must be adjusted for quasi geostrophy.

The QG form of Potential Vorticity is

q = ∇2_hΨ + ∂ ∂z f0 N2 ∂Ψ ∂z + f (1.90)

With f = f0 + βy. This expression of PV is exactly the quantity into

brackets appearing in equation (1.87), which represents indeed a balance of quasi geostrophic potential vorticity (here the constant value f0 of planetary

vorticity had been added, having no effects in (1.87) when derived). There are various ways to derive this theorem, one of these is starting from Ertel’s theorem. Otherwise, it is possible to derive it from the complete set of equation in its primitive form , which is somewhat elaborate. On the other hand, keeping in mind the scaling and expansion of our QG system, we can reach the same target much easier via Ertel’s theorem.

Ertel’s form of PV is given in equation (1.68), where θ is a scalar tracer and ω is the absolute vorticity. Ertel’s theorem in general form gives

ρD Dt  ω · ∇θ ρ  = ω · ∇(Gθ/ρ) + 1 ρ2∇θ · ∇ρ × ∇p + ∇θ · ( eF ) (1.91)

Where we should remember that eF is a term rising from the curl of non conservative term. This theorem states simply the conservation of Ertel’s PV in a frictionless flow (i.e eF = 0) that is conserving a tracer θ (i.e. Gθ = 0,

which is a function of density and pressure alone (i.e. ∇ρ × ∇p = 0) . In this condition is indeed

D Dt  ω · ∇θ ρ  = 0 (1.92)

In our case, we can choose the density as the tracer, θ = ρ = ρ0 + δρ

(base plus perturbation), which is conserved. Implementing the shallow-water and Boussinesq approximations, Ertel’s theorem in the general form (1.91) becomes

D Dt  (f + ζ) ∂ρ0 ∂z + ∂δρ ∂z  = ζ∂Gθ ∂z +  ∂ρ0 ∂z + ∂δρ ∂z  ( eF ) (1.93)

The baroclinicity term vanishes identically for θ = ρ. Expanding this equation with respect to all small parameters so that f = f0+βy , βy << f0,

ζ << f0 , δρ << ρ0, and taking w as small as well, we finally find, retaining

all terms up to the second order:

D Dt(f + ζ) ∂ρ0 ∂z + D Dtf0 ∂δρ ∂z + w ∂ ∂zf0 ∂ρ0 ∂z ≈ f0 ∂Gθ ∂z + ∂ρ0 ∂z ( eF ) (1.94)

With ζ = ∇2_hΨ , ∂ρ0/∂z = −N2/g. Then, using the hydrostatic balance

(1.32) and eliminating the vertical velocity w through (1.82) we finally obtain:

Dq Dt = −f0g ∂ ∂z Gθ N2 + ( eF ) (1.95)

Where , in our case of ( eF ) = 0 and Gθ = 0 , it is exactly the

same as (1.92) : qg-PV conservation law.

Note that, in contrast to Ertel’s PV (1.68) , the Quasi Geostrophic Po-tential Vorticity (1.90) is conserved following the geostrophic motion (on pressure surfaces).There are two important points that put in evidence the advantage coming from the use of qgPV

- The quasi geostrophic system of equation as expressed through qg-equation for momentum, mass continuity and density or buoyancy equa-tion (thermodynamic equaequa-tion), is a set of five equaequa-tions and involves three

scalar components of the velocity fields ( two geostrophic 0-order and one ageostrophic 1-order), plus pressure and density fields. Even if its closure can be reached as discussed, it still remains a quite complex calculation. It is possible to reduce this five-variables dependence to one that relies on a single variable from which all of the others can be derived. What we did introducing the qgPV is to include into one single equation, the qgPV con-servation law, the equations of motion , via stream function Ψ, which is the only prognostic variable that determines the entire flow at any time, as seen in the relations (1.88.a,b,c) and (1.89).

- The conserved (to Rossby order) quasi geostrophic PV is an elliptic fun-cion of the function Ψ. Under these circumstances, the spatial distribution of q can be inverted given certain boundary conditions, to yield the distribution of velocities and mass.

We have now in hand the core elements of the dynamics of quasi geostrophic flows: the twin principles of qgPV conservation and invertibility.

1.4.3

Summary of the QG equations via PV

We can finally resume the quasi geostrophic set of equations: QG equation written for quasi geostrophic potential vorticity via streamfunction, including friction at bottom, forcing and dissipating actions - equation (1.96) - with the boundary conditions for surface and bottom (1.97), and for the lateral walls (1.98) .

∂q

∂t + J (Ψ, q) = Fq+ Dq+ E[∇

2

∂b ∂t + J (Ψ, b) = Fb+ Db z = −H, 0 (1.97) ∂∆Γ ∂t = Z Z Fq+ Dq+ E[∇2Ψ]dxdy y = −L, L (1.98)

Where b is the buoyancy, defined as b = f ∂zΨ, q the interior

poten-tial vorticity, Ψ is the stream function, S(z) = N (z)_f is the stretching term and N(z) represents the mean stratification frequency; Fq and Fb are forcing

terms, acting respectively on PV and buoyancy, while Dq and Db are small

scale dissipation terms, again on PV and buoyancy.

The quantity ∆Γ = Γ(L, z)−Γ(−L, z) , appearing in the lateral boundary conditions (1.98), is the difference of circulation between the two walls, where circulation is Γ(y, z) =R ∂yΨdx .

The term E[∇2Ψ] has the form E[∇2Ψ] = −λδ(z + H)∇2Ψ and it retains the role of linear bottom drag with parameter λ, introduced to be a sink for the upscale energy cascade since it acts at large scales, as necessary. δ is the Dirac function. The time scale used to fix λ , namely the friction time scale Tf ric = λ−1, has been set at 50 days.

The surface PV qs is related to the surface buoyancy bsby qs= −S−2(0)bs

and likewise for the bottom PV qb = −S−2(−H)bb. The full QG model as

presented here might be considered a coupled interior-surface QG model. The few layers QG model consists in keeping interior PV and setting to zero both the surface and the bottom PV. While, surface QG model consists in the opposite.

The Jacobian between the streamfunction and the PV in the QG model equations has got the physical meaning that PV is advected along stream function isolines. The first advantage comes indeed from the use of the

Jacobian in equations (1.96) and (1.97), that reduces the computation of the quantity u · ∇(q) to the determinant of a 2x2 matrix, thanks to the relationship between geostrophic velocity and stream function.

Briefly, due to the definition of stream function and the property of the geostrophic velocity field to be divergence free, is clear that

J (Ψ, q) = ∂xΨ∂yq − ∂xq∂yΨ = v∂yq + u∂xq = u · ∇(q) (1.99)

In 2.1.1 we will explore this system in detail, since it is the core of the numerical code developed. Solving this system for q we can then invert the elliptic relation between q and Ψ, and once known the latter finally we go back to whole fields at any time through (1.88)-(1.89), which gives u, v, w, ρ via Ψ.

Chapter 2

Numerical simulations:

framework and routine

In this chapter we will describe the simulation set-up and the methodology.

The first section 2.1 is completely dedicated to introduce the simulation framework starting from a description of both the domain and the imple-mented model, and continuing with an outline of the computational proce-dure and the presentation of the advection schemes. The key concept of small scale implicit dissipation is also illustrated, as well as the reason why we explore the choice of including dissipation into the computation instead of having an explicit tunable term. It will be then explaned how this implicit dissipation is numerically computed. A part is dedicated to the numerical forcing and its design philosophy. The last paragraph is created to give the reader some basic information about two physical input parameters: buoy-ancy and mean stratification.

The second section 2.2 provides a summary of the simulation routine, tools and methodology, together with a summary of all the simulations car-ried out.

2.1

Simulations framework

2.1.1

The domain and the QG model

The domain chosen to run all the simulations is a reentrant channel on the beta-plane. The detailed explanation of the beta plane approximation has been given in section 1.1.2, to which we refer for further details. It is here just necessary to recall that the equations of motion can be represented on a tangent plane with Cartesian coordinates - which for small displacements do not differ much from the spherical ones- see relation (1.27). Furthermore, for small variations in latitude on this plane it is possible to mimic the behavior of the Coriolis parameter f (1.28) with a linear variation (1.29) in the merid-ional direction: the beta plane can be seen as a tangent plane where no big variations with latitude are present, or in other words it is a tangent plane extended for not a big range of latitudes, i.e. the mesoscale is sufficiently smaller than the global scale. Thanks to this choice, the domain geometry is very simple: it has a rectangular shape in both zonal and vertical direc-tion, as shown in figure 2.1. The dimensions are Lx = 768Km along the zonal, Ly = 2L = 1536Km along the meridional, and the vertical depth is H = 4Km.

The physical parameters have been chosen to be as similar as possible to those of the Southern Ocean (SO). It has been fixed a typical Coriolis parameter, given in terms of magnitude f = |f |, and a beta parameter β tipical of SO at mid latitude, as follows

Figure 2.1: Qualitative illustration of the channel domain. On the right side the full channel is represented, on the left the horizontal directions are illustrated, along with the Cardinal points letter to understand the orientation.

β = 2.56 · 10−11m−1s−1 (2.2)

Referring to the physical-mathematical environment, all the simulations are ranked amongst the Forced-Dissipative Problems (FDP).

The tests have been performed using a numerical QG model designed to solve the qg-equations written for Potential Vorticity. The big advantage is that it is possible to derive the velocity field solving just one equation for PV, instead of three equations for the velocity components. The detailed explanation and derivation of these equations, and how are they able to describe the mesoscale dynamics, has been treated in chapter one.

We can now write them as the system of equations, in continuous form, that represents the core of the QG model.

Defining the interior potential vorticity as

q = ∇2Ψ + ∂z(S−2∂zΨ) (2.3)

We can write the QG system to be solved

         ∂tq + J (Ψ, q) = Fq+ Dq+ E[∇2Ψ] ∂tb + J (Ψ, b) = Fb+ Db z = −H, 0 ∂t∆Γ =R R Fq+ Dq+ E[∇2Ψ]dxdy y = −L, L (2.4)

The first of (2.4) is the PV equation to be solved for the interior potential vorticity q, with the boundary conditions at the bottom and top - second

equation of (2.4)-, and for the lateral walls, third equation of (2.4). The detailed description of the terms has been given in section 1.4.3 (ho fatto una aggiunta a quella section in cui ne spiego il significato). The numerical structure, discretization and computation methodology will be given in next section 2.1.2.

We now just need to manipulate the first equation of (2.3) to obtain the implicit-dissipation formulation, which is directly resolved by our model. First of all, we recall that the term E[∇2_{Ψ] is non-zero only at the bottom,}

see [1] (riferimento ad un paper usato), and it does not affect the equations for the interior. We can write the PV interior equation as

∂tq + J (Ψ, q) − Dq = Fq (2.5)

And define a new Jacobian ˜J which includes the dissipative term

˜ J = J (Ψ, q) − Dq (2.6) Equation (2.5) becomes ∂tq + ˜J = Fq (2.7) Where we define Dq = − 1 2[ ˜J (Ψ, q) + ˜J (−Ψ, q)] (2.8)

J (Ψ, q) = 1

2[ ˜J (Ψ, q) − ˜J (−Ψ, q)] (2.9)

Equations (2.8-9) ensure that (2.6) is satisfied. Note that for the Jacobian terms a relationship of the following type holds

J (Ψ, q) = ∂xΨ∂yq − ∂xq∂yΨ = v∂yq + u∂xq = u · ∇(q) (2.10)

The formulation (2.7) means that the dissipation acts implicitly, instead of having an explicit dissipation term in the model. The term Dq is indeed

included into the Jacobian, the computation is more compact since Dq is

hidden into the advection scheme, which is applied to the ˜J .

In conclusion, we can write the first equation of the system (2.4), in form (2.7) and finally obtain the system (2.11).

         ∂tq + ˜J = Fq ∂tb + J (Ψ, b) = Fb+ Db z = −H, 0 ∂t∆Γ =R R Fq+ Dq+ E[∇2Ψ]dxdy y = −L, L (2.11)

The numerical model solves indeed the discretized version of system (2.11).

The advection part, represented by the full Jacobian, is computed in flux form exploiting a relationship of the type (2.10). The dissipation diagnostic is designed to be implicit. The details of this procedure will be shown in the next paragraph.

2.1.2

Computation methodology, discretization and

ad-vection schemes

We illustrate in this section: the conceptual procedure of the computation, the details of the discretization method and the advection schemes.

1. Conceptual procedure: The procedure we adopted for the computa-tion is a two stage methodology, based on the link between PV and stream function. The first stage is the advection stage: after initialization, the PV field is computed at next time step; in the second stage, the inversion stage, a new-step stream function is obtained.

This can be explained by the following schematic.

- Initialization: Set the initial condition at t0 = 0 on PV by choosing

a starting stream function. This is done via discretization of the equation (2.3), where we express the continuous (∇2_{+ ∂z(S}−2_{∂z)) with a discretized}

operator summarized as ∆q in the following.

Ψ(0) = Ψ0 ⇒ q0 = q(0) = ∆qΨ(0) (2.12)

At the initial time the discretized forcing is also zero, it is Fq(0) = −Λ(q(0)−

Qref)H, where Qref is set to be the initial pv itself. Details of the forcing

are given in section 2.1.5 .

- Advection stage: PV is computed at next time step t1 = t0 + ∆t

ex-ploiting the Jacobian relationship for advection. This stage relies on the discretized form of equation (2.7), that can be written as follows

Denoting with ˜J [Ψ0, q0] the discretized Jacobian.

- Inversion stage: the new stream function at t1 is obtained inverting the

potential vorticity via an elliptic operator ∆−1_q

Ψ(1) = ∆−1_q q(1) (2.14)

The boundary conditions for the surface and the bottom are the dis-cretized version of second) for the surface and the bottom and (2.11-third) along the walls. Of course from the knowledge of q(1) _{and Ψ}(1) _{it is}

possible to obtain the geostrophic field at the new time step. The schematic is summarized in figure (2.2) .

Splitting the computation as shown has revealed to be cheaper than using a single stage method. This conceptual procedure reproduces the main stages of the computation, but it is of course necessary to explain in detail how the advection term, i.e. Jacobian, and the time advancing are discretized and computed.

Figure 2.2: Conceptual computation schematic. Representation of one time step with the two-stage advection-inversion formulation. The core of our study is the advection stage, in which the advection numerical schemes directly compute the Jacobian, with various time integrators to perform the advection time advancing.

Figure 2.3: Left: finite volume grid in two dimensional space, where Qij is a cell

average value. Right: detail of the shaded grid cell and qualitative representation of Qij as a pointwise value.

2. Discretization and Advection schemes : for what concern the horizontal the variables are discretized on a staggered grid, of uniform spacing ∆x = ∆y = h, where ∆x = xi+1/2 − xi−1/2 and ∆y = yj+1/2 − yj−1/2, and the

advected PV is face-centered. The computation is based on high resolution finite volume methods: considering the given uniform grid in the xy directions and referring to figure (2.3) it is possible to define a value Q(n)_ij as the cell average over the (i,j) grid cell at a certain time step n as follows

Q(n)_ij ≈ 1 ∆x∆y Z j+1/2 j−1/2 Z x+1/2 x−1/2 q(x, y, t)dxdy (2.15)

determine how this cell average varies with time, and develop finite volume methods(FVM) based on numerical approximations of the fluxes at each cell edge. Rather than working with the cell average and the integral form, one could instead view Qij as a pointwise approximation at the cell center point

(xi, yj) as qualitatively illustrated in figure (2.3).

For what concern the z-direction the grid is vertically stretched according to ∆z0 = ∆x = ∆y, where ∆z0 = S(z)∆z. This implicitly sets the thickness ∆z of the vertical levels via S(z)∆z = ∆x. The level depths are computed iteratively as follows: first the grid uniformity is transformed into the equiv-alent form ∆zk∆ ¯bk = f2∆x2 , where ∆zk is the vertical thickness of a grid

cell and ∆ ¯bk is the associated buoyancy jump. Then, starting from equal

values, the ∆zk are modified iteratively until the previous equivalent form is

satisfied. Note anyway that the vertical direction does not cover a central role in this study, which is mainly focused on the horizontal QG turbulence. We briefly explained the vertical grid structure and we also point out that horizontal and vertical resolution are related each other thanks to the link be-tween the horizontal and vertical grid, as just expressed. For further details on the vertical non-geostrophic grid and field we refer to [1].

The core of our work is the computation of the advection stage. This is done computing the advection term in flux form - using various numerical schemes to compute the PV numerical fluxes - combined with various time integrators to perform the time advancement, as we here explain in detail.

The advection term is represented by the discretized Jacobian ˜J [Ψ, q] ap-pearing in the advection stage in the schematic of figure (2.2). This term is computed exploiting a discretized version of equation (2.10), which is repre-sented by the following equation (2.16).

L[q] = (F_i+1/2,jx − Fx i−1/2,j+ F y i,j+1/2− F y i,j−1/2)/h (2.16)

Figure 2.4: Localization of the cell centered potential vorticity qij, and velocities

u_i±1/2,j, v_i,j±1/2 which are instead located at cell edges. Where ∆x = x_i+1/2− xi−1/2 and ∆y = yj+1/2− yj−1/2 The grid cell size is shaded and the velocity

points are spotted with small green points

Where the terms F_i+1/2,jx , F_i−1/2,jx , F_i,j+1/2y and F_i,j−1/2y are the numerical fluxes of potential vorticity through the cell interfaces. They reads

F_i+1/2,jx = ui+1/2,jq˜i+1/2,j

F_i−1/2,jx = ui−1/2,jq˜i−1/2,j

F_i,j+1/2y = vi,j+1/2q˜i,j+1/2

F_i,j−1/2y = vi,j−1/2q˜i,j−1/2

(2.17)

where ˜q is an interpolation of qI at velocity point. In figure (2.4) the

location on the grid cell of the centered advected potential vorticity as well as of the velocity points at grid location yi,j±1/2 and xi±1/2,j are shown.

The formulation (2.16) for the advective term is the same for all the tested schemes, while each scheme differs from the others for the numerical fluxes representation.

In other words, the advection scheme sensitivity is performed testing var-ious numerical methods for express all the (2.17), thus (2.16), along with various time schemes to perform the time step. When a flux is needed at a fractional timestep, the velocity is extrapolated at this time using a three points interpolation (at tn_{, t}n−1 _{and t}n−2_).

In particular we tested 3rd order and 5th order Upwind schemes, a WENO 5th order and a MUSCL 2nd order with three different flux limiters. The schemes are summarized in Table 2.1 , where spatial flux discretizations are combined with various time integrators.