IMPACTS OF SIGMA COORDINATES ON THE EULER...

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IMPACTS OF SIGMA COORDINATES ON THE EULERAND NAVIER-STOKES EQUATIONS USING

CONTINUOUS/DISCONTINUOUS GALERKINMETHODS

Sean L. GibbonsCaptain, United States Air Force

B.S. Materials Science, United States Air Force Academy, 2003

Submitted in partial fulfillment of therequirements for the degrees of

MASTER OF SCIENCE IN METEOROLOGYMASTER OF SCIENCE IN APPLIED MATHEMATICS

from the

NAVAL POSTGRADUATE SCHOOLMarch 2009

Author: Sean L. Gibbons

Approved by: Francis Giraldo, Co-Advisor

Maj Tony Eckel, Co-Advisor

Philip Durkee, ChairmanDepartment of Meteorology

Carlos Borges, ChairmanDepartment of Applied Mathematics

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iv

ABSTRACT

In this thesis, ... Three test cases are analyzed: A rising thermal bubble, a

linear hydrostatic mountain, and a linear nonhydrostatic mountain. The methods

will be outlined for the transformation of two sets (set 1 the non-conservative form

using Exner pressure, momentum, and potential temperature; set 2 the conservative

form using density, momentum, and potential temperature) of the x-z Navier-Stokes

equations to x-σz...

The same transformation for sigma-z vertical coordinates used by COAMPS,

WRF and other mature mesoscale models will be applied to the two sets of the

Navier-Stokes equations of interest. After applying the sigma-z coordinates, the dis-

cretization method of choice employs continuous/discontinuous Galerkin techniques.

The existing code is in Fortran and all of the necessary modifications will also be

made in Fortran.

After the modifications have been made to the model, three test cases will be

run: rising thermal bubble, linear hydrostatic mountain, and linear non-hydrostatic

mountain. The numerical solutions will then be evaluated against either other model

solutions (case 1) or the analytic approximations (case 2 and case 3) using root mean

squared error and L2 error norms. The resultant data will then be compared to the

unmodified solutions. The initial data for the test case is pre-generated by the source

code, maintaining uniform initial conditions from which both coordinate systems

numerical solutions can be compared.

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TABLE OF CONTENTS

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A. GOVERNING EQUATIONS . . . . . . . . . . . . . . . . . . . . 3

1. Equation Set 1: Non-conservative . . . . . . . . . . . . . 3

2. Equation Set 2: Non-conservative . . . . . . . . . . . . . 3

B. X-Z TO X-σZ COORDINATE SYSTEM TRANSFORM . . . . 4

1. Gal-Chen and Somerville . . . . . . . . . . . . . . . . . . 4

2. Basic Transformation Machinery . . . . . . . . . . . . . . 6

3. Transformation Functions . . . . . . . . . . . . . . . . . 9

C. SPATIAL DISCRETIZATION . . . . . . . . . . . . . . . . . . . 10

D. TEMPORAL DISCRETIZATION RK4 . . . . . . . . . . . . . . 13

III. APPLIED COORDINATE TRANSFORMS . . . . . . . . . . . . 15

A. EQUATION SET 1 . . . . . . . . . . . . . . . . . . . . . . . . . 15

1. Perturbation Method . . . . . . . . . . . . . . . . . . . . 15

2. Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3. Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 19

4. Application of the Galerkin Statement . . . . . . . . . . 20

B. EQUATION SET 2 . . . . . . . . . . . . . . . . . . . . . . . . . 21

1. Perturbation Method . . . . . . . . . . . . . . . . . . . . 21

2. Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3. Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 25

4. Application of the Galerkin Statement . . . . . . . . . . 26

IV. TEST CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A. CASE 1: RISING THERMAL BUBBLE . . . . . . . . . . . . . 29

B. CASE 2: LINEAR HYDROSTATIC MOUNTAIN . . . . . . . . 29

C. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN . . . . . 31

vii

V. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A. OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B. CASE 1: RISING THERMAL BUBBLE . . . . . . . . . . . . . 33

1. Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2. Comparison and Conclusions . . . . . . . . . . . . . . . . 33

C. CASE 2: LINEAR HYDROSTATIC MOUNTAIN . . . . . . . . 34

1. Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


D. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN . . . . . 34

1. Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


VI. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . 35

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

INITIAL DISTRIBUTION LIST . . . . . . . . . . . . . . . . . . . . . . 39

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LIST OF FIGURES

1. The stability of the explicit leapfrog time-integrator. Figure a) has no

time-filter, while figure b) has a time-filter weight of ǫ=.05. The solid

lines represent the physical solutions while the dashed lines represent

the computational modes. . . . . . . . . . . . . . . . . . . . . . . . . . 30

ix


x

LIST OF TABLES

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ACKNOWLEDGMENTS

I would like to thank my advisors, Francis X. Giraldo and Major Tony Eckel,

without whom this thesis would not have been possible. I would also like to thank

the Naval Postgraduate School, the Meteorology and Mathematics departments at

NPS, The Air Force Institute of Technology, The United States Air Force, and The

United States Navy.

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I. INTRODUCTION

word [1] word [2] blah [3] blah [4]

This is a NWP topic that examines coordinate systems and continuous or

discontinuous Galerkin methods in relation to the Navier-Stokes equations. The pro-

posed title of the thesis is: Impacts of Sigma Coordinates on the Euler and Navier-

Stokes Equations using Continuous or Discontinuous Galerkin Methods.

There are multiple sets of governing equations that can be used to describe

atmospheric flow. The Navier-Stokes equations, along with their variations, form the

most widely used and accepted sets of equations for numerically resolving atmospheric

flow. Two specific formulations of the equation sets will be the focus of this study. In

order to use discontinuous Galerkin methods to solve the Navier-Stokes equations, the

equations have been written in conservation form. Since there is no conservative form

of the first set of equations, continuous Galerkin methods have to be used to solve

them. Dr. Francis X. Giraldo, implementing continuous/discontinuous Galerkin tech-

niques, developed a 2-D (x-z slice) mesoscale model using Non-Hydrostatic Equations

(Euler and Navier-Stokes Equations). The original construct used z for the vertical

coordinates. In this study, the current formulation of the Navier-Stokes equations will

be transformed using sigma-z vertical coordinates to test their impacts on resolving

atmospheric motion in a continuous/discontinuous Galerkin framework.

Will implementing sigma-z coordinates significantly improve or diminish the

solution of the Navier-Stokes equations over x-z coordinates when using continu-

ous/discontinuous Galerkin methods?

1


2

II. BACKGROUND

A. GOVERNING EQUATIONS1. Equation Set 1: Non-conservative

the Pressure Tendency Equation:

∂π

∂t+ ~u · ∇π +

R

cvπ∇ · ~u = 0 (2.1)

the Momentum Equations (µ = 0):

∂~u

∂t+ ~u · ∇~u+ cpθ∇π = −g~k (2.2)

the Thermodynamic Energy Equation (µ = 0):

∂θ

∂t+ ~u · ∇θ = 0 (2.3)

2. Equation Set 2: Non-conservative

the Mass Equation:

∂ρ

∂t+ ∇ · (ρ~u) = 0 (2.4)

the Momentum Equations:

∂~u

∂t+ ~u · ∇~u+

1

ρ∇P = −g~k (2.5)

3

the Thermodynamic Energy Equation:

∂θ

∂t+ ~u · ∇θ = 0 (2.6)

B. X-Z TO X-σZ COORDINATE SYSTEM TRANSFORM

1. Gal-Chen and Somerville

In 1975, Gal-Chen and Somerville, took the anelasic approximation of the

Navier-Stokes Equation (in the cartesian form) and transformed the coordinated sys-

tem to sigma-z coordinates. The initial equations consisted of:

the Continuity Equation:

(p0uj),j = 0.


(

∂

∂t

)

(ρ0ui) + (ρ0u

iuj),j = −(δijp′),j +δi3ρ′g + τ ij ,j .


(

δ

δt

)

(ρ0θ′) + (ρ0θ

′uj),j = Hj,j .

the Eddy Viscosity:

τ ij = ρ0KM

[

eij −

(

2δij

δii

)

(uk,k )

]

.

the Eddy Diffusion:

H i = ρKHδij

(

∂θ′

∂xj

)

.

4

the Variable Eddy Viscosity:

KM = (k∆)2|Def |[

1 −(

KH

KM

)

(Ri)′]1/2

.

the Variable Heat Diffusion:

KH/KM = constant = 1/Pr.

the Modified Richardson Number:

(Ri)′ =

Ri : |θ′| ≤ 10−3θ0

Ri− (|∂p′

∂x||∂θ′

∂x| + |∂p′

∂y||∂θ′

∂y|)/(|ρθ′|(Def)2) : otherwise

the Richardson Number:

Ri =(

g

θ0

)

(

∂θ′

∂z

)

/(Def)2.

the (Def)2:

(Def)2 = 0.5τ ijeij/(ρ0KM) =1

2eijeij − (2/δij)(uk,k )2

and eij:

eij =

(

∂ui

∂xj

)

+

(

∂uj

∂xi

)

a. Transformation Functions

The set of transformations used by Gal-Chen and Somerville were:

~x = x, y = y, z =H(z − zs)

(H − zs)

∂z

∂x=∂zs

∂x

z −H

H − zs

,∂z

∂y=∂zs

∂y

z −H

H − zs

,∂z

∂z=

H

H − zs

5

u

v

w

=

1, 0, 0

0, 1, 0

∂zs

∂xz−HH−zs

, ∂zs

∂yz−HH−zs

, z−HH−zs

u

v

w

with the inverse transformations:

x = x, y = y, z = [z(H − zs)

H] + zs

u

v

w

=

1, 0, 0

0, 1, 0

−∂zs

∂xz−H

H, −∂zs

∂yz−H

H, H−zs

H

u

v

w

2. Basic Transformation Machinery

This section will outline the basic equation used to set up the Navier-Stokes

Equations for transformation. The concept used was the total differential:

dx =∂x

∂xdx+

∂x

∂zdz

dσz =∂σz

∂xdx+

∂σz

∂zdz

6

where the ’ notation indicates the transformed variable. The two total derivatives

can then be written as a system of equations:

dx

dz

=

∂x∂x

∂x∂z

∂σz

∂x∂σz

∂z

dx

dz

Using vector notation ( ~over bar) the above system can be simplified to:

d~x = Jd~x

where J is the Jacobian. Using the definition for velocity (~u):

d~x

dt= ~u

the transformation becomes:

~u = J~u

The inverse transform for velocity can be written as:

~u = J−1~u

The transform was for the gradient operator (∇) is:

∂

∂x=

∂

∂x

∂x

∂x+

∂

∂σz

∂σz

∂x

∂

∂z=

∂

∂x

∂x

∂z+

∂

∂σz

∂σz

∂z

7

The gradient can then be written as a matrix:

∂∂x

∂∂z

=

∂x∂x

∂σz

∂x

∂x∂z

∂σz

∂z

∂∂x

∂∂σz

Using vector notation the above system can be simplified to:

∇ = JT ∇ (2.7)

the last basic definition used for the transformation is:

(AB)T = BTAT (2.8)

Proof:

(AB)T =

a1 a2

a3 a4

b1 b2

b3 b4

T

=

a1b1 + a2b3 a1b2 + a2b4

a3b1 + a4b3 a3b2 + a4b4

T

=

a1b1 + a2b3 a3b1 + a4b3

a1b2 + a2b4 a3b2 + a4b4

BTAT =

b1 b3

b2 b4

a1 a3

a2 a4

=

a1b1 + a2b3 a3b1 + a4b3

a1b2 + a2b4 a3b2 + a4b4

8

3. Transformation Functions

The set of transformations are:

x = x, y = y, σz =H(z − zs)

(H − zs)(2.9)

∂σz

∂x=∂zs

∂x

σz −H

H − zs

,∂σz

∂y=∂zs

∂y

σz −H

H − zs

,∂σz

∂z=

H

H − zs

(2.10)

J =

1, 0

∂zs

∂xσz−HH−zs

, HH−zs

(2.11)

JT =

1, ∂zs

∂xσz−HH−zs

0, HH−zs

(2.12)

J−1 =

1, 0

−∂zs

∂xσz−H

H, H−zs

H

(2.13)

(J−1)T =

1, −∂zs

∂xσz−H

H

0, H−zs

H

(2.14)

u

w

=

1, 0

∂zs

∂xσz−HH−zs

, HH−zs

u

w

(2.15)

9

with the inverse transformations:

x = x, y = y, z = [σz(H − zs)

H] + zs (2.16)

u

w

=

1, 0

−∂zs

∂xσz−H

H, H−zs

H

u

w

(2.17)

C. SPATIAL DISCRETIZATION

To construct the problem we will first consider the we will first consider the

generalized 2-D hyperbolic-elliptic PDE:

∂q

∂t+ ~u · ∇q = ν∇2q

where q = q(~x, t), ~u = ~u(~x), ~x = (x, z)T , and ν is the viscosity coefficient. Using

Galerkin machinery, qN and ~u where approximated using basis function expansion:

qN(~x, t) =MN∑

j=1

ψj(~x)qj(t)

~uN(~x, t) =MN∑

j=1

ψj(~x)~uj(t)

10

where ψj is the Lagrange polynomial basis functions. The approximations for qN

and ~uN were then substituted into the PDE, multiplied by a test function, ψI , and

integrated in the global domain, Ω, to get (weak integral form):

∫

ΩψI∂qN∂t

dΩ +∫

ΩψI(~u · ∇qN)dΩ = ν

∫

ΩψI∇

2qNdΩ ∀Ψ ∈ H1

Instead of solving the global problem directly, 2-D local basis functions where con-

structed and then direct stiffness summation (DSS) was used to construct the global

problem. The 2D local basis functions are defined as:

ψi(ξ, η) = hj(ξ) ⊗ hk(η)

where:

hj(ξ) =N∏

l = 0

l 6= j

(

ξ − ξlξj − ξl

)

Additionally, in order to construct the basis functions and transition between physical

and computational space requires known of the metric terms:

∂ξ

∂x=

1

|J |

∂z

∂η,∂ξ

∂z=

−1

|J |

∂x

∂η

∂η

∂x=

−1

|J |

∂z

∂ξ,∂η

∂z=

1

|J |

∂x

∂ξ

11

|J | =∂x

∂ξ

∂z

∂η−∂x

∂η

∂z

∂ξ

converting the above using integration by parts (IBP):

∫

Ωe

ψi∇2qNdΩe =

∫

Ωe

∇ · (ψi∇qN )dΩe −∫

Ωe

∇ψi · ∇(qN )dΩe

then using divergence theorem:

∫

Ωe

ψi∇2qNdΩe =

∫

Γe

~n · (ψi∇qN )dΓe −∫

Ωe

∇ψi · ∇(qN)dΩe

and subbing the result back into the PDE produced:

∫

Ωe

ψi∂qN∂t

dΩe +∫

Ωe

ψi(~u · ∇qN)dΩe = ν∫

Γe

~n · (ψi∇qN)dΓe − ν∫

Ωe

∇ψi · ∇(qN )dΩe

Subbing for the summation approximation for qN and ~uN yields:

∫

Ωe

ψi

MN∑

j=1

ψj∂qj∂t

dΩe +∫

Ωe

ψi

(

MN∑

k=1

ψk~uk

)

·

MN∑

j=1

∇ψjqj

dΩe

= ν∫

Γe

ψi~n ·

MN∑

j=1

∇ψjqj

dΓe − ν∫

Ωe

∇ψi ·

MN∑

j=1

∇ψjqj

dΩe

12

The resulting matrix problem is:

M(e)ij

∂qj∂t

+ A(e)ij (~u)qj = B

(e)ij qj − L

(e)ij qj

D. TEMPORAL DISCRETIZATION RK4

13


14

III. APPLIED COORDINATE

TRANSFORMS

A. EQUATION SET 11. Perturbation Method

For this section both π and θ with be broken into two components mean (

π and θ) and their perturbations (π′ and θ′). the Pressure Tendency Equation 2.1

becomes:

∂(π + π′)

∂t+ ~u · ∇(π + π′) +

R

cv(π + π′)∇ · ~u = 0

∂π′

∂t+ ~u · ∇π′ + w

∂π

∂z+R

cv(π + π′)∇ · ~u = 0 (3.1)

the Momentum Equations 2.2 becomes:

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)∇(π + π′) = −g~k

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)

[(

∂π

∂x+∂π

∂z

)

+

(

∂π′

∂x+∂π′

∂z

)]

= −g~k

dπ

dz= −

g

cpθ

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)

[

−g

cpθ~k +

(

∂π′

∂x+∂π′

∂z

)]

= −g~k

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)

(

∂π′

∂x+∂π′

∂z

)

− g~k − gθ′

θ~k = −g~k

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)∇π′ = g

θ′

θ~k (3.2)

15

the Thermodynamic Energy Equation 2.3 becomes:

∂(θ + θ′)

∂t+ ~u · ∇(θ + θ′) = 0

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂z= 0 (3.3)

2. Transform

Using the basic machinery prescribed in equations 2.7 - 2.15 the set of the non-

conservative Navier-Stokes (equations 3.1 - 3.3) was prepared for transformation:

Applying the machinery to the Pressure Tendency Equation yields:

∂π′

∂t+ ~uT∇π′ + w

∂π

∂z+R

cv(π + π′)∇ · ~u = 0

∂π′

∂t+ (J−1~u)T (JT ∇)π′ + w

∂π

∂z

∂σz

∂σz

+R

cv(π + π′)(JT ∇)T (J−1~u) = 0

∂π′

∂t+ (~u)T (J−1)T (JT )(∇)π′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂π

∂σz

∂σz

∂z

+R

cv(π + π′)(∇)T (JT )T (J−1)(~u) = 0

∂π′

∂t+ (~u)T (∇)π′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂π

∂σz

(

H

H − zs

)

+R

cv(π + π′)(∇)T (~u) = 0

16

∂π′

∂t+ ~u · ∇π′ − u

(

σz −H

H − zs

)

∂zs

∂x

∂π

∂σz+ w

∂π

∂σz+R

cv(π + π′)∇ · ~u = 0

∂π′

∂t+ ~u · ∇π′ − u

(

σz −H

H − zs

)

∂zs

∂σz

∂π

∂x+ w

∂π

∂σz+R

cv(π + π′)∇ · ~u = 0

∂π′

∂t+ ~u · ∇π′ + w

∂π

∂σz+R

cv(π + π′)∇ · ~u = 0 (3.4)

Applying the machinery to equation 3.2 yields:

∂~u

∂t+ ~uT∇~u+ cp(θ + θ′)∇π′ = g

θ′

θ~k

∂~u

∂t+ (J−1~u)T (JT ∇)~u+ cp(θ + θ′)(JT ∇)π′ = g

θ′

θ~k

∂~u

∂t+ (~u)T (J−1)T (JT )(∇)~u+ cp(θ + θ′)(JT ∇)π′ = g

θ′

θ~k

∂~u

∂t+ (~u)T (∇)~u+ cp(θ + θ′)(JT ∇)π′ = g

θ′

θ~k

17

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)(JT ∇)π′ = g

θ′

θ~k (3.5)

Applying the machinery to equation 3.3 yields:

∂θ′

∂t+ ~uT∇θ′ + w

∂θ

∂z= 0

∂θ′

∂t+ (J−1~u)T (JT ∇)θ′ + w

∂θ

∂z

∂σz

∂σz

= 0

∂θ′

∂t+ (~u)T (J−1)T (JT )(∇)θ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂θ

∂σz

∂σz

∂z= 0

∂θ′

∂t+ (~u)T (∇)θ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂θ

∂σz

(

H

H − zs

)

= 0

∂θ′

∂t+ ~u · ∇θ′ − u

σz −H

H − zs

∂zs

∂x

∂θ

∂σz+ w

∂θ

∂σz= 0

∂θ′

∂t+ ~u · ∇θ′ − u

σz −H

H − zs

∂zs

∂σz

∂θ

∂x+ w

∂θ

∂σz= 0

18

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂σz= 0 (3.6)

3. Decomposition

The Pressure Tendency Equation 3.4:

∂π′

∂t+ ~u · ∇π′ + w

∂π

∂σz+R

cv(π + π′)∇ · ~u = 0

decomposed becomes:

∂π′

∂t+

[

u∂π′

∂x+ w

∂π′

∂σz

]

+ w∂π

∂σz+R

cv(π + π′)

[

∂u

∂x+∂w

∂σz

]

= 0 (3.7)

The Momentum Equation 3.5:

∂~u

∂t+ ~u · ∇~u+ cp(θ + θ′)(JT ∇)π′ = g

θ′

θ~k

decomposed becomes:

∂u

∂t+

[

u∂u

∂x+ w

∂u

∂σz

]

+ cp(θ + θ′)

[

∂π′

∂x+

(

∂zs

∂x

σz −H

H − zs

)

∂π′

∂σz

]

= 0 (3.8)

and

19

∂w

∂t+

[

u∂w

∂x+ w

∂w

∂σz

]

+ cp(θ + θ′)

[

(

H

H − zs

)

∂π′

∂σz

]

= gθ′

θ(3.9)

The Thermodynamic Energy Equation 3.6:

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂σz

= 0

decomposed becomes:

∂θ′

∂t+

[

u∂θ′

∂x+ w

∂θ′

∂σz

]

+ w∂θ

∂σz= 0 (3.10)

4. Application of the Galerkin Statement

∂π′

∂t+

[

u∂π′

∂x+ w

∂π′

∂σz

]

+ w∂π

∂σz+R

cv(π + π′)

[

∂u

∂x+∂w

∂σz

]

= 0

∂u

∂t+

[

u∂u

∂x+ w

∂u

∂σz

]

+ cp(θ + θ′)

[

∂π′

∂x+

(

∂zs

∂x

σz −H

H − zs

)

∂π′

∂σz

]

= 0

∂w

∂t+

[

u∂w

∂x+ w

∂w

∂σz

]

+ cp(θ + θ′)

[

(

H

H − zs

)

∂π′

∂σz

]

= gθ′

θ

20

∂θ′

∂t+

[

u∂θ′

∂x+ w

∂θ′

∂σz

]

+ w∂θ

∂σz= 0

B. EQUATION SET 2

1. Perturbation Method

∂ρ

∂t+ ∇ · (ρ~u) = 0

∂ρ

∂t+ ~u · ∇ρ+ ρ∇ · ~u = 0

∂(ρ+ ρ′)

∂t+ ~u · ∇(ρ+ ρ′) + (ρ+ ρ′)∇ · ~u = 0

∂ρ′

∂t+ ~u · ∇ρ′ + w

∂ρ

∂z+ (ρ+ ρ′)∇ · ~u = 0 (3.11)


∂~u

∂t+ ~u · ∇~u+

1

ρ∇P = −g~k

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇(P + P ′) = −g~k

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇P ′ +

1

(ρ+ ρ′)

∂P

∂z~k = −g~k

21

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇P ′ −

ρg

(ρ+ ρ′)~k = −g~k

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇P ′ +

ρ′g

(ρ+ ρ′)~k = 0

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇P ′ = −

ρ′g

(ρ+ ρ′)~k (3.12)


∂θ

∂t+ ~u · ∇θ = 0

∂(θ + θ′)

∂t+ ~u · ∇(θ + θ′) = 0

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂z= 0 (3.13)

2. Transform

Using the basic machinery prescribed in equations 2.7 - 2.15 the set of the

non-conservative Navier-Stokes (equations 3.11 - 3.13) was prepared for transforma-

tion where the Mass Equation 3.11 becomes:

∂ρ′

∂t+ ~u · ∇ρ′ + w

∂ρ

∂z+ (ρ+ ρ′)∇ · ~u = 0

∂ρ′

∂t+ ~uT∇ρ′ + w

∂ρ

∂z+ (ρ+ ρ′)∇T~u = 0

22

∂ρ′

∂t+ (J−1~u)T (JT ∇)ρ′ + w

∂ρ

∂z

∂σz

∂σz+ (ρ+ ρ′)(JT ∇)TJ−1~u = 0

∂ρ′

∂t+ (~u)T (J−1)T (JT ∇)ρ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂ρ

∂σz

∂σz

∂z+ (ρ+ ρ′)(∇)T (JT )TJ−1~u = 0

∂ρ′

∂t+ (~u)T ∇ρ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂ρ

∂σz

(

H

H − zs

)

+ (ρ+ ρ′)(∇)T ~u = 0

∂ρ′

∂t+ ~u · ∇ρ′ + w

∂ρ

∂σz

+ (ρ+ ρ′)∇ · ~u = 0 (3.14)

the Momentum Equations 3.12 becomes:

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)∇P ′ = −

ρ′g

(ρ+ ρ′)~k

∂~u

∂t+ ~uT∇~u+

1

(ρ+ ρ′)∇P ′ = −

ρ′g

(ρ+ ρ′)~k

∂~u

∂t+ (J−1~u)TJT ∇~u+

1

(ρ+ ρ′)JT ∇P ′ = −

ρ′g

(ρ+ ρ′)~k

23

∂~u

∂t+ ~u

T(J−1)TJT ∇~u+

1

(ρ+ ρ′)JT ∇P ′ = −

ρ′g

(ρ+ ρ′)~k

∂~u

∂t+ ~u

T∇~u+

1

(ρ+ ρ′)JT ∇P ′ = −

ρ′g

(ρ+ ρ′)~k

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)JT ∇P ′ = −

ρ′g

(ρ+ ρ′)~k (3.15)

the Thermodynamic Energy Equation 3.13 becomes:

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂z= 0

∂θ′

∂t+ ~uT∇θ′ + w

∂θ

∂z= 0

∂θ′

∂t+ (J−1~u)TJT ∇θ′ + w

∂θ

∂z

∂σz

∂σz

= 0

∂θ′

∂t+ (~u)T (J−1)TJT ∇θ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂θ

∂σz

∂σz

∂z= 0

24

∂θ′

∂t+ (~u)T ∇θ′ +

(

−u∂zs

∂x

σz −H

H+ w

H − zs

H

)

∂θ

∂σz

(

H

H − zs

)

= 0

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂σz= 0 (3.16)

3. Decomposition

The Mass Equation 3.14:

∂ρ′

∂t+ ~u · ∇ρ′ + w

∂ρ

∂σz

+ (ρ+ ρ′)∇ · ~u = 0

decomposed becomes:

∂ρ′

∂t+

[

u∂ρ′

∂σz+ w

∂ρ′

∂σz

]

+ w∂ρ

∂σz+ (ρ+ ρ′)

[

∂u

∂x+∂w

∂σz

]

= 0 (3.17)

The Momentum Equation 3.15:

∂~u

∂t+ ~u · ∇~u+

1

(ρ+ ρ′)JT ∇P ′ = −

ρ′g

(ρ+ ρ′)~k

decomposed becomes:

∂u

∂t+

[

u∂u

∂x+ w

∂u

∂σz

]

+1

(ρ+ ρ′)

[

∂P ′

∂x+

(

∂zs

∂x

σz −H

H − zs

)

∂P ′

∂σz

]

= 0 (3.18)

25

and

∂w

∂t+

[

u∂w

∂x+ w

∂w

∂σz

]

+1

(ρ+ ρ′)

[

(

H

H − zs

)

∂P ′

∂σz

]

= −ρ′g

(ρ+ ρ′)(3.19)

The Thermodynamic Energy Equation 3.16:

∂θ′

∂t+ ~u · ∇θ′ + w

∂θ

∂σz= 0

decomposed becomes:

∂θ′

∂t+

[

u∂θ′

∂x+ w

∂θ′

∂σz

]

+ w∂θ

∂σz= 0 (3.20)

4. Application of the Galerkin Statement

∂ρ′

∂t+

(

u

[

∂ρ′

∂x+

(

∂zs

∂x

σz −H

H − zs

)

∂ρ′

∂σz

]

+ w

[

(

H

H − zs

)

∂ρ′

∂σz

])

+w∂ρ

∂σz+ (ρ+ ρ′)

[

∂u

∂x+∂w

∂σz

]

= 0

∂u

∂t+

[

u∂u

∂x+ w

∂u

∂σz

]

+1

(ρ+ ρ′)

[

∂P ′

∂x+

(

∂zs

∂x

σz −H

H − zs

)

∂P ′

∂σz

]

= 0

26

∂w

∂t+

[

u∂w

∂x+ w

∂w

∂σz

]

+1

(ρ+ ρ′)

[

(

H

H − zs

)

∂P ′

∂σz

]

= −ρ′g

(ρ+ ρ′)

∂θ′

∂t+

[

u∂θ′

∂x+ w

∂θ′

∂σz

]

+ w∂θ

∂σz

= 0

27


28

IV. TEST CASES

A. CASE 1: RISING THERMAL BUBBLE

For this test case there is no terrain:

zsurf = 0, ∀x

which lead to:

∂zsurf

∂x= 0, ∀x

making the transformation of the coordinate system (x-zσ) reduced to x-z.

B. CASE 2: LINEAR HYDROSTATIC MOUNTAIN

For this test case the terrain is represented by:

zsurf =hc

(

1 +(

x−xcac

)2) , ∀x

which lead to:

zsurf = hc

(

1 +(

x− xc

ac

)2)

−1

zsurf = hc

[

(ac)2 + (x2 − 2(xc)(x) + (xc)2)

(ac)2

]

−1

29

A

x

z

0 500 10000

200

400

600

800

1000B

x

z

0 500 10000

200

400

600

800

1000

C

x

z

0 500 10000

200

400

600

800

1000

a)

Figure 1. The stability of the explicit leapfrog time-integrator. Figure a) has notime-filter, while figure b) has a time-filter weight of ǫ=.05. The solid lines representthe physical solutions while the dashed lines represent the computational modes.

zsurf = (hc)(ac)2[

(ac)2 + x2 − 2(xc)(x) + (xc)2]

−1

∂zsurf

∂x=

(−1)(hc)(ac)2(2x− 2(xc))

[(ac)2 + x2 − 2(xc)(x) + (xc)2]2

∂zsurf

∂x=

(−2)(hc)(ac)2(x− xc)

[(ac)2 + (x− xc)2]2, ∀x

30

C. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN

31


32

V. RESULTS

A. OVERVIEWB. CASE 1: RISING THERMAL BUBBLE

1. Accuracy

2. Comparison and Conclusions

33

C. CASE 2: LINEAR HYDROSTATIC MOUNTAIN1. Accuracy

2. Comparison and ConclusionsD. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN

1. Accuracy2. Comparison and Conclusions

34

VI. CONCLUSIONS AND

RECOMMENDATIONS

This study will aid in determining the usefulness of applying a specific coordi-

nate system in the future, when developing meteorological and oceanographic models

for the US Naval Research Laboratory (NRL) by constituents at the Naval Postgrad-

uate School. In addition, the successful conversion of the non-hydrostatic x-z models

to x-sigma-z will allow for the straightforward extension of these models to global

non-hydrostatic models.

35


36

LIST OF REFERENCES

[1] M. Restelli F.X. Giraldo. A study of continuous and discontinuous galerkin meth-ods for the navierstokes equations in nonhydrostatic mesoscale atmospheric mod-eling: Equation sets and test cases. Journal of Computational Physics, 227:3849–3877, 2008.

[2] C.J. Somerville T. Gal-Chen. On the use of a coordinate transformation forthe solution of the navier-stokes equations. Journal of Computational Physics,17:209–228, 1975.

[3] R.M. Hodur. The naval research laboratorys coupled ocean/atmosphere mesoscaleprediction system (coamps). Monthly Weather Review, 125:1414–1430, 1997.

[4] W.C. Skamarock J.B. Klemp J. Dudhia D.O. Gill D.M. Baker W. Wang J.G.Powers. A description of the advanced research wrf version 2. NCAR TechnicalNote NCART/TN-468+STR, 2007.

37


38

INITIAL DISTRIBUTION LIST

1. Defense Technical Information CenterFt. Belvoir, Virginia

2. Library, Code 52Naval Postgraduate SchoolMonterey, California

39

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