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Adjoint data assimilation in an open ocean barotropic quasi-geostrophic model 1993
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Title | Adjoint data assimilation in an open ocean barotropic quasi-geostrophic model |
Creator |
Bailey, David A. |
Date Created | 2008-09-18 |
Date Issued | 2008-09-18 |
Date | 1993 |
Description | A barotropic quasi-geostrophic ocean model with open boundaries was used to model a system of mid-ocean eddies. A simplified adjoint assimilation scheme was tested to see if sparse velocity data could be assimilated into the model at regular intervals. In between the times for data assimilation, the model was integrated forward in time with an Orlanski radiating boundary condition. This assimilation scheme was tested with several model runs, illustrating the changes arising from using different eddy sizes, different density of available data, and different numerical model parameters. This scheme was also compared with a bicubic interpolation scheme. During data assimilation, the resulting velocity field was generally more accurate than that obtained by interpolation alone. However, the Orlanski radiating boundary condition was not very effective in suppressing the growth of errors after data assimilation. |
Extent | 2405087 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2008-09-18 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053277 |
Degree |
Master of Science - MSc |
Program |
Oceanography |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/2271 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0053277/source |
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ADJOINT DATA ASSIMILATION IN AN OPEN OCEAN BAROTROPIC QUASI-GEOSTROPHIC MODEL By David Anthony Bailey B.Math., University of Waterloo, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF OCEANOGRAPHY INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1993 © David Anthony Bailey, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature Department of Oceistx■ (3rap The University of British Columbia Vancouver, Canada Date ^oct- 9) 19cta DE-6 (2/88) Abstract A barotropic quasi-geostrophic ocean model with open boundaries was used to model a system of mid-ocean eddies. A simplified adjoint assimilation scheme was tested to see if sparse velocity data could be assimilated into the model at regular intervals. In between the times for data assimilation, the model was integrated forward in time with an Orlanski radiating boundary condition. This assimilation scheme was tested with several model runs, illustrating the changes arising from using different eddy sizes, different density of available data, and different numerical model parameters. This scheme was also compared with a bicubic interpolation scheme. During data assimilation, the resulting velocity field was generally more accurate than that obtained by interpolation alone. However, the Orlanski radiating boundary condition was not very effective in suppressing the growth of errors after data assimilation. ii Table of Contents Abstract^ ii Table of Contents^ iii List of Tables iv List of Figures^ v Acknowledgements^ vii 1 Introduction^ 1 2 Method^ 4 2.1 Problem Definition ^4 2.2 Model ^7 2.2.1 Analytical Model ^7 2.2.2 Numerical Model ^ 10 2.3 Adjoint Model ^ 13 3 Results^ 17 3.1 Description of Parameters ^ 17 3.2 Test Cases and Results 21 4 Conclusions^ 48 Bibliography^ 50 iii List of Tables 2.1 Analytical model parameters ^ 9 2.2 Numerical model notation 10 3.1 Reference ocean parameters ^ 18 3.2 Assimilation model variables 18 iv List of Figures 2.1 Nested domains ^5 2.2 Illustration of the computational boundary ^12 3.1 Two different grid samples ^ 19 3.2 Initial reference ocean - distance scale 200 km ^22 3.3 Initial reference ocean - distance scale 400 km ^ 23 3.4 Nondimensional total kinetic energy (TKE) for the period of the experi- ments with a distance scale of 200 km. ^ 24 3.5 Nondimensional total kinetic energy (TKE) for the period of the experi- ments with a distance scale of 400 km ^ 24 3.6 Test 1 - Day 387 ^ 25 3.7 Test 1 - Day 513 26 3.8 Test 1 - RMS Errors ^ 27 3.9 Test 2 - Day 387 29 3.10 Test 2 - Day 513 ^ 30 3.11 Test 2 - RMS Errors 31 3.12 Test 2 - RMS interpolated velocity error^ 31 3.13 Test 3 - Day 387 ^ 32 3.14 Test 3 - Day 513 33 3.15 Test 3 - RMS Errors ^ 34 3.16 Test 3 - RMS interpolated velocity error^ 34 3.17 Test 4 - Day 387 ^ 35 3.18 Test 4 - Day 513 ^ 36 3.19 Test 4 - RMS Errors 37 3.20 Test 4 - RMS interpolated velocity error ^37 3.21 Test 5 - Day 387 ^ 38 3.22 Test 5 - Day 513 39 3.23 Test 5 - RMS Errors ^ 40 3.24 Test 5 - RMS interpolated velocity error^ 40 3.25 Test 6 - Day 387 ^ 41 3.26 Test 6 - Day 508 42 3.27 Test 6 - RMS Errors ^ 43 3.28 Test 6 - RMS interpolated velocity error^ 43 3.29 Test 7 - Day 387 ^ 45 3.30 Test 7 - Day 508 46 3.31 Test 7 - RMS Errors ^ 47 vi Acknowledgements I would like to thank Dr. W. W. Hsieh for his guidance and support over the period of my study, and during the writing of this thesis. Thank-you to Dr. B. T. R. Wetton and Dr. S. E. Allen for encouraging remarks and comments leading to the final draft of this thesis. Thank-you also to Dr. J. Zou and Mr. W. Lee for their assistance in the background research. I would also like to mention various graduate students and staff in the Departments of Mathematics and Oceanography for fielding a number of my questions. Special thanks to Mr. J. M. Stockie for his support and the informative late night discussions. Financial support from the Natural Sciences and Engineering Research Council of Canada in the form of a postgraduate scholarship is gratefully acknowledged. vii Chapter 1 Introduction One problem with a numerical model is to determine suitable boundary conditions to obtain a well-posed problem. Normally one only has enough computer resources to model a part of the world ocean, hence there has to be an artificial condition at the edge of the domain. A closed boundary in a numerical model occurs where the edge of the domain intersects a piece of land or a wall. The conditions for a solid wall (or "no-slip") boundary have been well established in numerical ocean models for some time. (See [1, 3, 18, 19] for examples.) The question of an open boundary, where the domain ends in the open ocean say, is very difficult because of the need for information to propagate freely out of the domain without affecting the interior solution. Also there is no way to determine the disturbances propagating from outside the numerical domain into the solution region. The classical idea for an open boundary was to specify an analytical or exact solution at inflow points and extrapolate the solution from the interior at the outflow points (see Charney et al. [4].) It was discussed by Miller and Bennett [9] that this scheme does not give a well-posed problem, except for very special cases. There is an intense vorticity build-up at the boundary, but with a sufficient amount of friction or viscosity the scheme can be used fairly accurately. The Harvard open ocean model of Robinson and Haidvogel [14] used this type of boundary condition and included a filtering mechanism to deal with the build-up of energy. Another class of open boundary conditions are the "radiation" schemes considered by Orlanski [11], Engquist and Majda [5], Raymond and Kuo [13], and others. These 1 Chapter 1. Introduction^ 2 type of conditions consider a local wave equation near the boundary with an apparent phase velocity, controlling the way in which disturbances exit the domain. The authors generally differ in the way they consider this apparent phase velocity and the order of the derivatives in the equations. Another side of physical oceanography is the collection of observed data from satellite imagery, current meters, drifters, and so on. The vastness of the ocean almost always ensures a sparse data set. Often these data are also incomplete due to cloud coverage and weather conditions. Mathematical techniques such as bicubic interpolation and objective analysis can be used to "interpolate" the data thus giving a more complete picture. Mathematical optimization and control theory can also be used in the form of "adjoint data assimilation." Data assimilation has been used in meteorology for a number of years for operational forecasting. Thacker [15, 16, 17] considers the idea of adjoint data assimilation in physical oceanography. Data assimilation is a wide term for an area including the use of observed data in mathematical models. It could include injection of data into the model, Kalman filtering and variational or adjoint data assimilation. Interpolation of data would involve curve- fitting the data-points with no real use of the physics of the ocean. The interpolation between data-points can be very good, but is only as accurate as the observations. Data injection is the technique in which a mathematical model is run and model values are actually replaced by observed data values. This causes discontinuities in the solution and generates much noise. The most sophisticated data assimilation techniques are Kalman filtering (see [2] Chapter 3) and adjoint data assimilation. In adjoint assimilation, a cost function is defined to give the error or misfit between the observed data and the corresponding model values. Certain parameter values must then be determined so that the cost function is minimized. This way the model solution is smoothed and "nudged" closer to the observed data. Chapter 1. Introduction^ 3 Adjoint data assimilation can be used in a number of ways. It is used by meteorologists in a forecasting sense. Observed wind data are used to correct the model at certain times to a given atmospheric state, then the model is allowed to run again. Data assimilation is also used in what is called an "inverse" model, to determine certain parameters which gave rise to a particular state. An example would be to determine the friction or wind parameters which cause a mid-ocean eddy stream. Another way to use adjoint data assimilation is in the completion of sparse observed data as described earlier. Zou and Hsieh [20] considered the idea of using adjoint data assimilation to determine the boundary values at one open boundary of a two-dimensional domain. They considered satellite sea-surface elevations (streamfunction) with a simple model. This combined with a nested domain concept used by Robinson and Haidvogel [14] led to the idea of using adjoint data assimilation to determine the boundary values of a completely open two- dimensional domain to complete a set of drifter or current meter velocity data. This was the idea of this thesis, to use a simple ocean model and adjoint data assimilation to complete a set of sparse data. Note that a two-dimensional rectangular domain with open boundaries (no inflow condition) on all four sides by itself is a not a well-posed problem, but with data assimilation the problem should be better posed. Chapter 2 describes the methodology of the thesis: the problem definition, analytical and numerical model, and the adjoint technique. Chapter 3 considers seven cases which test the effect of the parameters in the model and compare the data assimilation scheme to a standard bicubic interpolation routine. Chapter 4 is the summary of the results and conclusions. Chapter 2 Method 2.1 Problem Definition The background research of the thesis involved two main steps: first, to modify a simpli- fied adjoint method as implemented by Zou and Hsieh [20] to use an open ocean model with a nested domain concept used by Robinson and Haidvogel [14]; and secondly, to change the algorithm to assimilate velocity data from the interior of the innermost region of the reference ocean instead of streamfunction data. Robinson and Haidvogel [14] performed an experiment using nested domains (see Figure 2.1.) In that experiment they used a simple ocean model with a 1000 by 1000 kilometer, rectangular domain (called the reference ocean) which was sampled to obtain a time series of streamfunction and vorticity values. The streamfunction (&) and the vorticity (C) were defined as: _ u _ (2.1) — ay ,v ax av au =^— (2.2) where u is the velocity with u, v the velocity components in the cartesian x (eastward) and y (northward) directions respectively. The sampled data from an inner 500 by 500 kilometer region centred in the reference ocean were used as initial and boundary conditions for a model domain of the same size as the inner region of the reference ocean. In the adjoint data assimilation work by Thacker [15, 16, 17], he defined a cost function J, which was a measure of the error between some observed quantities d, and 4 Chapter 2. Method^ 5 Figure 2.1: Nested domains their numerical model counterparts m. J (m — d)TA(m — d),^ (2.3) where A was the inverse of an error covariance matrix of the observations, and the superscript T referred to the transpose of a vector quantity in this case. If the observed data values were assumed to be independent and their errors uncorrelated, then A was simply the identity matrix I. The problem was then to minimize the cost function with respect to certain control variables and subject to the constraining physical equations. Zou and Hsieh [20] considered a two-dimensional, rectangular domain with three closed boundaries and one open boundary. A time series of streamfunction (sea surface elevation) was saved from a reference model run. This time series was sampled to obtain interior streamfunction information for the model run. An adjoint method was then solved for the streamfunction on the southern boundary and a correction to the interior vorticity. Note that full adjoint assimilation would have involved a number of forward time integrations of the numerical model, and backward integrations of an adjoint model Chapter 2. Method^ 6 (see [15, 17].) This was computationally expensive, so Zou and Hsieh only did the assimilation in space at selected time steps to save computational work. Zou and Hsieh [20] defined the following cost function: J(03, 60 = (0 — ib)T (0 — 0), (2.4) which was an implicit function of the control variables O s , the streamfunction values along the southern (open) boundary of the domain and an interior vorticity correction 5 discussed later. The ';i) were reference model streamfunction values and & were the assimilation model counterparts of the reference streamfunction. The problem was to minimize the cost function J with respect to the control variables, and subject to a barotropic quasi-geostrophic model (next section) as the constraint. The combination of the nested domain idea with the adjoint scheme above became the basis of the problem to follow. Velocity was the assimilated quantity instead of streamfunction. The velocity data could come from drifter or other current measurements in the real world, but were generated by the reference ocean for these experiments. So the cost function in this work was defined as: JON , 60 = (u — fi)T (u — fi), (2.5) minimized over the control variables Ob, the streamfunction along the whole boundary, and the interior vorticity correction as used by Zou and Hsieh and also subject to a barotropic quasi-geostrophic model. The model counterpart u, of the observed velocity ii, was obtained from the derivatives of the streamfunction (see equations (2.1).) Chapter 2. Method^ 7 2.2 Model 2.2.1 Analytical Model The mathematical model chosen for the experiments was a flat-bottomed "barotropic quasi-geostrophic" (QG) model. Quasi-geostrophy refers to an almost geostrophic sit- uation in which the velocity basically follows the pressure contours (i.e. the pressure gradient balances the Coriolis force, see [6]) on a tangent or beta-plane approximation to the Earth's surface. A barotropic model describes a two-dimensional, vertically inte- grated ocean. In this work, the vorticity-streamfunction formulation was used as opposed to the velocity-pressure formulation in the primitive equation model. The linear and nonlinear, inviscid (no friction) barotropic equations have sinusoidal plane wave (Rossby wave) solutions. A superposition of two Rossby waves is not however a solution to the nonlinear equations. (See [7] section 38.) When this superposition of two Rossby waves was used to force the reference domain with the full nonlinear equations (including linear bottom friction), it developed a nonlinear eddy field which was said to be an example of the "real" ocean. This was the idea used by Robinson and Haidvogel [14] in which they chose wavelength and period values from a wave-fit of data from the Mid- Ocean Dynamics Experiment (MODE) in the Atlantic Ocean. (The wave-fit calculations were done by McWilliams and Flierl [8].) One form of the barotropic vorticity equation with horizontal and lateral viscosity and no wind-stress (see [1, 3, 18]) was given as: a(^a at + J(0, 0 13 ax+ — = —KbV 20 + KhV40,^(2.6) o where, 2^a2^a2 V = - + aX 2 ay2 a4^a2 a2^a4V4 = - + 2— — -I-ax4^ax2 ay2 ay4 , (2.7) (2.8) Chapter 2. Method^ 8 Kb was the bottom friction coefficient, K h was the horizontal viscosity coefficient, and J was the Jacobian given by the formula: ao a( azk a(,7(0, C) = ax -a-y- - ay ax (2.9) On a beta-plane (tangent plane) at mid-latitudes, the Coriolis parameter f is assumed to be a linear function of the north-south cartesian co-ordinate y, (see [6] Chapter 12): f = fo + fiy,^ (2.10) where f = 25/ sin 0, is the Coriolis parameter with 11 Pze, 7.3 x 10 -5 s' , the Earth's angular velocity, and 0 the latitude. At mid-latitudes (0 0 = 45°), the beta-plane parameter is given as, p = (211/R e ) cos 00 '',-' 2.0 x10 -11m's-1 and the zeroth order Coriolis parameter is fo = 212 sin 00 r.--, 1.0 x 10 -4s". The dimensions of the rectangular domain were defined to be 0 < x, y < L. (L was 1000 km for the reference model, and 500 km for the assimilation model.) The model was non-dimensionalized by a choice of scaling: x = 'Xd, y = 'yd, t = 4/34- 1 , 0 = li)(Vod), and C = ((Void) where d was a length scale (typical eddy size) and Vo was a velocity scale. So using the scaled variables and omitting the carets the non-dimensional barotropic quasi-geostrophic equations for the reference and model domain were given by: a(^ao 79t- + R3(0,0 + -a; = -Ebv 20 + Ehv40,^(2.11) 02,0 _ c ,^(2.12) where equation (2.12) follows directly from the definition of vorticity and streamfunction (see equations (2.1) and (2.2).) The equation parameters are summarized in Table 2.1. The Rossby wave boundary conditions for the reference ocean were given as: 2,b = Ai COS(kiX + liy — Wit + 01) + A2 COS(k2X + 12Y — W2t + 02),^(2.13) Chapter 2. Method^ 9 VoR = Oda 1'6 = fld Kh eh = #(13 Rossby number Ekman bottom friction Laplace friction Table 2.1: Analytical model parameters where, Ai =^Ki = VIC? -I- q, Ki were the wavelengths and wave vector magnitudes, T^27r= wi (2.14) (2.15) were the periods, A i the two wave amplitudes, ki and /i the east-west and north-south wave numbers, wi the frequencies, and cb i the phase shifts. The model ocean required four open boundaries, i.e. four boundaries at which dis- turbances from the interior propagated freely out of the domain without affecting the interior of the domain. This relied on using some type of extrapolation from the inte- rior of the domain to the boundary. One formula for this, described by Orlanski as the Sommerfeld radiation condition [11] was given by the equation: -at + = ° (2.16) where was a dependent model quantity from the interior of the domain (such as zi) or () and co was the apparent phase velocity at the boundary. The phase velocity was a quantity which had to be estimated by a local distance and time scale near the boundary, described in the next section. Yi+1,j ^—2 n-1^—4 n-1 2At 20x^ebV^EhV 0i,j n+1 Ci>i = R,1*(0 7.a •i,3 (2.17) Chapter 2. Method^ 10 2.2.2 Numerical Model The type of numerical method typically used in physical oceanography is a centred fi- nite difference scheme. This type of numerical model has been used successfully by Arakawa [1], Bryan [3], Veronis [19] and others. The finite difference approximations to equations 2.11 and 2.12 were: (2.18) where 3* is the numerical Jacobian formulation due to Arakawa [1]. V 2 and V4 are numerical approximations to the standard Laplacian derivatives where: V 2 n+1^+ It-2-11,i 4. 7/ ,z1-41 +^40,7v. (Ax)2 in the case when Ax = Ay, and V 4 has a similar, but more complicated form. (The numerical model notations are summarized in Table 2.2.) (2.19) I J N Ax Ay At i,j,n ti ((iAx,jAy,nAt) P.:111)(iAx,j0y,nAt) number of grid points in the x direction number of grid points in the y direction number of time steps grid spacing in the x direction grid spacing in the y direction time step indices for space and time the finite difference approximation to the vorticity the finite difference approximation to the streamfunction Table 2.2: Numerical model notation The boundary conditions for the streamfunction in the finite difference reference model were just a straight discretization of the analytical conditions: "Ki,3 ^A l cos(ki (iAx) ii(jAY)— wi((n +1)At) cbi) (07bi-1 -- 6-1 on-2) (07b1-1^— 2071:1D At ' _ot ox Ax (2.23) Chapter 2. Method^ 11 + A2 cos(k2(iLx) i2(jAY) w2((n + 1).6d) + 02).^(2.20) for i, j on the boundary. To calculate the vorticity, an extra computational boundary was introduced where the streamfunction was extrapolated from the interior, ^1 7c = 206^yob-1. ^ (2.21) where the subscript c indicated the computational boundary, b indicated the physical boundary, and b —1 one grid point in from the boundary (see Figure 2.2.) The vorticity at inflow points was given by the discretization of the analytical solution, ^— '4(14. + cos(ki(iLx)^Ay) — wi((n + 1)At) 01) — A2 (14 + cos(k 2 (i6a) /2(jAy) — w2 ((n 1)At) + 02), (2.22) for i, j at inflow boundary points, and at the outflow points was calculated from the standard finite difference discretization of the Poisson equation (2.18). The boundary conditions for the model runs were the discretization of the Sommerfeld condition (2.16). The phase velocity near the boundary was estimated to be — and was limited to the region 0 < Co < t with this resulted in, (2.24) as the values of the phase velocity. So the value of a variable on the boundary was given by: 14:4-1 11 (t)C 01 on-1 + 2 (E)C on [1 + (DC _I b^+ (t)Col (2.25) Chapter 2. Method^ 12 Figure 2.2: Illustration of the computational boundary. The subscripts c, b, and b — 1 refer to the computational boundary, physical boundary, and one grid point to the interior from the physical boundary respectively. Chapter 2. Method^ 13 where 0 would be streamfunction or vorticity. The general solution method was to obtain the vorticity at step one (C i, ) from the initial data (z/4 ^093 ) and a forward Euler time step i.e., ^ = RHS of (2.17),^ (2.26) At then solve for the streamfunction at step one (01-i ) using equation (2.18) which was solved with a standard numerical Poisson equation package, then continue at the next step with equation (2.17). The accuracy of the finite difference scheme was 0((Ax) 2 (At) 2 ) because of the use of the centred differences in space and centred difference or "leap-frog" in time (with a lag for the diffusion terms.) The reference model was stable for a sufficiently small time step and appropriate choice of e l, and eh . In the test cases of the following chapter, Eb = 8.0 x 10' and Eh = 2.0 x 10'. These values were considered fairly small, but reduced the build-up of noise in the reference model vorticity. 2.3 Adjoint Model The first step of the adjoint method was to define a cost function: =^\-• Er( n-1-1^^ n-1-1 )2 +^\ 21—2^p,q^P,9^kuP,9^uP,4 )P q (2.27) where p and q ranged over the points at which data were available and the model velocities were given by the centred finite difference approximations to equations (2.1) as follows: opri,i-q +1^opn:1-q^n+^opn+-1-11,^opnil-11 n+1^ 4 UP,4^2Ay , vp , q 2Ax • (2.28) Thacker [15] introduced a penalty or smoothing term in cases where there was a lack of data. This type of term was necessary for all but the first of the test cases of the next Chapter 2. Method^ 14 chapter, to ensure stability of the numerical model. The cost function with the addition of a penalty term took the following form: a 1-2 J-2J =_ + _2 E E^C:3-.1-+11^4(7V )2^(2.29) 1=3 j=3 where = cen1 j1 is the corrected vorticity, and a was the smoothing parameter. The arbitrary constant a could only be determined by "trial and error", but was found to lie between zero (no smoothing) and one (almost complete smoothing.) Given the penalized cost function (2.29), the optimal boundary control (OBC) problem was defined to be: min J(Vkl l , (5C1). 214,4C with k, 1 on the boundary, i, j in the interior and subject to the constraints, 2 9 /,/. /71-1 = T'10 (2.30) on the boundary,^(2.31) where CP came from the potential vorticity equation (2.17). The 1085 control variables for this problem were the boundary streamfunction values /kb and a vorticity correction 45C at all the interior points of the numerical grid. This vorticity correction was used by Zou and Hsieh [20] to deal with the vorticity at assimilation steps without actually assimilating vorticity data, nor carrying the vorticity prognostic equation (2.17) in the Lagrangian L in (2.32). To solve the problem, the method of Lagrange multipliers was used. The Lagrangian L, which includes the constraints, was defined as: 1-2 J-2 L j+EE 1411 1 (V214.7f1 (27.ti S(nt1)/ ^(2.32) i=3 j=3 where itZt i were the Lagrange multipliers. The Lagrange multipliers were solved for by computing the gradients of L with respect to the streamfunction at the interior points aJ k, 1 at data points, 0^otherwise,1 -1=72 n+1^8071:1-1V PO = (2.34 ) Chapter 2. Method^ 15 and setting the gradients to zero as follows: aL ^+ _77214_71_1= 0, 0017,1-1^azkiZt i which could be re-written as: (2.33) where, OJ oozy. — (uk,/-1 — 2Ay (vk-i,/ — j4--1,1) 2Ax + (uk,i+i — 2Ay (vk+1,1 - 1)k+1,1) •2Ax (2.35) The gradients with respect to the control variables, 0 6, and SC, of L were as follows: aL _ dJ aori — dob+1 aJ^ „n-1-1 Pb-1 aq+1 AxAy , (2.36) where ^- had a similar form to equation (2.35), and.h rl aL _ dJ n+1 0W:7 1 ) d(6Ci,J ) = aF(C:3±1) _ zj-1 where, (2.37) F( 71.2:1-1)zd —^ j++12^(in++21,i^-(7-27.1"1-214 9(/-n-1-1 .^) `n+1,j-1^'32-1,j-I-1 -1- x-1,j-1) Ci7+11,3 ) + 20 Ci7 E1 '^— 8 ( 3++^31 1 +-11 + C++11, 3 (2.38) (from Thacker [15] page 37), for all i, j in the interior of the numerical grid. Notice that the gradients of L with respect to the control variables (equations (2.36) and (2.37)) could be written explicitly in terms of the Lagrange multipliers. This made Chapter 2. Method^ 16 the solution method easier. Given the gradients and the Lagrange function, the prob- lem was solved using the NAG (Numerical Algorithms Group) Fortran library routine E04DGF. The NAG routine minimizes an unconstrained nonlinear function using a pre- conditioned, limited memory quasi-Newton conjugate gradient method. (See [10] for further details.) So the solution method is as follows: 1. Set up model parameters. 2. Obtain vorticity at step 1 (0 •) from initial vorticity and streamfunction (C 9, and ozpi ,a ), and using forward Euler step (equation (2.26).) 3. Obtain OLi from equation (2.18). 4. Advance vorticity according to potential vorticity equation (2.17) until next assim- ilation step. 5. If data are available for assimilation - solve OBC problem for obn+1 , (bn+1 and interior vorticity correction WI' 6. If there is no data available - specify 7,br i , and Cbi+ 1 according to a radiation boundary condition (2.25). 7. Solve Poisson equation (2.31) with (5(1:1 -1 = 0 in the case with no data. 8. Continue at step 4, but use a forward Euler step if assimilation was just done. The idea was to ignore the solution from the step before the assimilation. Chapter 3 Results 3.1 Description of Parameters The main parameters which influenced the pattern of the reference ocean were the periods (T1 , T2 ) and wavelengths (A 1 , A2 ) of the Rossby waves. The distance scale (d) was chosen to be the distance across a typical eddy, controlled by the choice of the wavelengths and periods of the two Rossby waves. This distance scale, given a fixed typical mid-ocean velocity scale (V0 ) of 10 cm/s was an indication of the Rossby number (R). Note that the choice of distance scale gave a non-dimensional basin length and width greater than one. The variables in the assimilation model included smoothing (a) and frequency of assimilation in time (n a ) and space (i a , ja ). The smoothing parameter was an arbitrary constant found by a number of experiments not included in the next section. There was no systematic method for obtaining a other than "trial and error". Figure 3.1 shows two types of data sampling strategies in space. Tables 3.1 and 3.2 summarize the parameters in dimensional units used in the experiments of the next section. The time step and grid spacing in the model were fixed. The inner model grid contained 33 by 33 grid points and the larger reference domain was 65 by 65 grid points. These values corresponded to a grid spacing of about 15 kilometers in both directions for both grids. The time step was one day (dimensional units) which gave good stability in the reference model, but the assimilation model required smoothing (i.e. non-zero 17 Chapter 3. Results^ 18 Expt. d (km) Ti. (days) T2 (days) A i (km) A2 (km) R 1. 200 161 129 171 291 0.125 2. 200 161 129 171 291 0.125 3. 200 161 129 171 291 0.125 4. 400 80.5 59.5 342 582 0.03125 5. 400 80.5 59.5 342 582 0.03125 6. 200 161 129 171 291 0.125 7. 200 161 129 171 291 0.125 Table 3.1: Reference ocean parameters Expt. a na (days) is ja 1. 0.0 5 1 1 2. 0.01 5 2 2 3. 0.002 5 3 3 4. 0.04 5 2 2 5. 0.04 5 3 3 6. 0.1 10 2 2 7. 0.05 5* 2 2 Table 3.2: Assimilation model variables. (* - assimilation done every five steps, but data only available every ten steps. Linear interpolation used to obtain data for odd steps.) Chapter 3. Results^ 19 Figure 3.1: Two different grid samples, at every (a) second, (i. = ja = 2) and (b) third, (i. = ja = 3) node in both directions. (The dots indicate grid points where data are sampled.) Chapter 3. Results^ 20 a in (2.29)) to keep the scheme stable. The difficulty with reducing the time step in the assimilation model was that it increased the number of time steps between data assimilations. If the assimilation was to be done at a realistic frequency in time, anything less than a time step of one day and assimilation on every fifth day would not have been viable. The reference ocean parameters (see Table 3.1) changed the overall picture of the domain, making the eddies larger or smaller, and more or less nonlinear. The assimilation model variables (see Table 3.2) affect the accuracy of the simulation. The following section gives seven test cases which show how the parameters and variables influence the solution. The error between the reference and the model oceans were measured in a root mean square sense (RMS) as seen in the following formulas: .,1f f(Om — lkr ) 2 dA I 2 (3.1)RMS( ) = [ f j(07.) 2 dA if f((m — 6.) 2dAl 2 (3.2)RMS(C) = [ f f(0 2dA f f {(um — ur ) 2 + (vm — vr)2}dAl^ (3.3) 2 RMS (u) — { f f {(ur ) 2 + (vr ) 2 1c/A where the RMS streamfunction error was considered over the whole domain (including the boundary) and the RMS vorticity and velocity errors were only considered on the interior. The vorticity and velocity on the boundaries were ignored since they were artificial and only used to advance the model for the nonassimilation steps. Chapter 3. Results^ 21 3.2 Test Cases and Results Given an initial pattern and boundary conditions, the reference ocean was left to "spin- up" for 258 days which was representative of twice the period of the oscillations in the Rossby waves. (The initial and boundary streamfunction contours for the two distance scales of the experiments are given by Figures 3.2(a) and 3.3(a).) The streamfunction fields at day 258 (Figures 3.2(b) and 3.3(b)) were then used as initial and boundary conditions for the reference ocean to be compared to the assimilation experiment. The experiments lasted for another 258 days. The RMS velocity error from the assimilation results were compared to the RMS error of a velocity field interpolated by a standard bicubic interpolation routine from the Numerical Recipes package (see [12] pages 118-120.) In most cases, the Numerical Recipes routine gave an overall RMS error less than that of the assimilation model. At assimilation steps, in the cases with data at every second node, the assimilation RMS error drops below the the interpolated RMS error. The RMS error in the numerical model, between assimilation steps grew more rapidly than the RMS error from linear interpolation in time. A comparison to an interpolated streamfunction was not done in test case one since it had been assumed that data were available at every interior grid point, thus there was no need to interpolate. Any RMS error values which were greater than unity had been set to unity. In the case of streamfunction, an error of greater than unity usually indicated that the general pattern of the contours was similar to the reference ocean, but the actual values were shifted up or down by a constant value. Since it was the velocity (a derivative of the streamfunction) which was assimilated, it was expected that the streamfunction could only be solved for up to an arbitrary constant. In four of the test cases, the RMS streamfunction error followed some type of a large Chapter 3. Results^ 22 Figure 3.2: Reference ocean streamfunction for distance scale of 200 km at (a) day zero and (b) day 258. Chapter 3. Results^ 23 Figure 3.3: Reference ocean streamfunction for distance scale of 400 km at (a) day zero and (b) day 258. Chapter 3. Results^ 24 oscillation. Perhaps this was caused by the fact that streamfunction was only solved up to an arbitrary constant as mentioned earlier. There seemed to be a small oscillation in the overall energy of the reference oceans (see Figures 3.4 and 3.5), but not really corresponding to the oscillations seen in the RMS streamfunction errors. 100.0 80.0 60.0w Y I- 40,0 20.0 0.0 250.0 350.0 450.0 550.0 Days Figure 3.4: Nondimensional total kinetic energy (TKE) for the period of the experiments with a distance scale of 200 km. 100.0 80.0 60.0w Y 1-- 40.0 20.0 0.0 250.0 350.0 450.0 550.0 Days Figure 3.5: Nondimensional total kinetic energy (TKE) for the period of the experiments with a distance scale of 400 km. The first test case was designed to demonstrate how the model reacts to the assimi- lation scheme. The theory was that if data were sampled at every point in the interior, the optimal-control problem could be solved with no need for smoothing. Figures 3.6 Chapter 3. Results^ 25 and 3.7 demonstrate the direct comparison of streamfunction and vorticity contours at two different time steps. Figure 3.8 shows the RMS errors of the velocity, vorticity, en- ergy, and streamfunction. The error rises during the nonassimilation steps, then drops rapidly at the assimilation steps as expected for vorticity, velocity, and energy. The RMS streamfunction (Figure 3.8(d)) error shows the strange oscillatory behaviour mentioned earlier. Figure 3.6: Test 1 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) Chapter 3. Results^ 26 Figure 3.7: Test 1 - Day 513: a comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. U' 0.40 cc 0.80 2 0.60 0.00 250.0 350.0 450.0 550.0 (a) 0.20 1.00 11,1'i^I 1,111;1'1^I 1.00 0.80 0.60 0.40 0.20 0 . 00 ^ 250.0 350.0 450.0 550.0 Days C" 0.40 cr 0.20 0.00 250.0 350.0 450.0 550.0 Days 0.80 L2 0.60 w 1.00 Chapter 3. Results^ 27 1.00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 (b) (c ) ^ (d) Figure 3.8: Test 1 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). Chapter 3. Results^ 28 The second test case considers a spatial sampling of every second node in both direc- tions, thus giving data at 256 points out of the 961 interior numerical grid points. There was some noise evident in the vorticity contours (see Figures 3.9(d) and 3.10(d).) The streamfunction contours (see Figures 3.9(b) and 3.10(b)) had a pattern very close to the reference contours and actual values were nearly identical. There was a bit of noise near the boundaries in the streamfunction contours. The smoothing parameter allowed the scheme to converge as well as reducing some of the noise in the vorticity. The error in the vorticity for the assimilation model was generally below forty percent, and hovered around five percent in the velocity. There was a strange oscillation in the RMS vorticity and streamfunction error, but both were fairly well behaved. When the data were only available at every third node in both directions (giving 121 sampled points), it was much more difficult for the model to converge. A smoothing parameter could be found so that the RMS errors did not diverge in the experiment time. There was much more noise in the vorticity contours than in the cases of every second node. The RMS velocity error never went above thirty percent, but it seemed to be increasing and never dropped below the interpolated velocity error. (Compare Figures 3.15(a) and 3.16.) Even the RMS energy error seemed to be increasing. The streamfunction contours were fairly smooth, but the actual values differed significantly from the reference streamfunction contours. The model vorticity still resembled the reference vorticity despite the increase in noise. (See Figures 3.13(d) and 3.14(d).) In test four, where the eddies are of a larger scale than in the first three test cases, the RMS error in the vorticity dropped rapidly to about twenty percent as the influence of the assimilation scheme became dominant. Even the streamfunction RMS error seemed to be declining as the experiment progressed. This trend did not seem to correspond to the TKE of the reference ocean (see Figure 3.5.) The velocity error varies uniformly around ten percent and dropped below the interpolated error at assimilation steps (compare Chapter 3. Results^ 29 Figure 3.9: Test 2 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) Chapter 3. Results^ 30 Figure 3.10: Test 2 - Day 513: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. 1 .00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 1 .00 0.80 2 0.60 Cl) 0.40 0.20 0.00 250.0 350.0 450.0 550.0 1 .0 0.8 0.6 0.4 0.2 0.0 250.0 350.0 450.0 550.0 ••••11111111111•••1111111111111111111••• 1.00 0.80 2 0.60 C 0.40 0.20 0.00 250.0 350.0 450.0 550.0 Days (c) Chapter 3. Results^ 31 (a) ^ ( b ) 1 . 00 0.80 0.60 0.40 0.20 0 . 00 ^m.1111111[1.111[11.10 250.0 350.0 450.0 550.0 Days (d) Figure 3.11: Test 2 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). Days Figure 3.12: Test 2 - RMS interpolated velocity error. Chapter 3. Results^ 32 Figure 3.13: Test 3 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) Chapter 3. Results^ 33 Figure 3.14: Test 3 - Day 513: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. Chapter 3. L Results 1.00 0.80 - 2Lw 0.60 (.) 0.40 0.20 0.00 1111111111111tt 11111111111111 34 III I III I 1 I I I I I I I Ili I I 1111111 1 1.00 0.80 0.60 0.40 0.20 0.00 If 250.0 350.0 450.0 550.0 (a) 1.00 0.80 L 2 0.60Lw C 0.40mm 0.20 ( c ) 250.0 350.0 450.0 550.0 (b) 1.00 0.80 0.60 0.40 0.20 0.00^FIIIIIIIIIIIIIIIIIIImilillo 250.0 350.0 450.0 550.0 Days (d) 0.00 250.0 350.0 450.0 550.0 Days Figure 3.15: Test 3 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). (Errors greater than unity truncated at unity.) 1.0 0.8 0.6 0.4 0.2 0.0^hilliiiihmilliiIIIIIIIIIII 250.0 350.0 450.0 550.0 Days Figure 3.16: Test 3 - RMS interpolated velocity error. Chapter 3. Results^ 35 Figures 3.19(a) and 3.20), but with much more variance than in previous cases. The patterns between the reference ocean and assimilation model contours were very close as well as the actual values. Also there seemed less noise in the vorticity, just a small amount near the boundaries. (See Figures 3.17 and 3.18.) Figure 3.17: Test 4 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) As in test case three, having data at every third node was difficult even with the larger eddy sizes of test case five. The RMS errors were very large in the streamfunction and vorticity, but there was at least more consistency in the trends. The RMS velocity error Chapter 3. Results^ 36 Figure 3.18: Test 4 - Day 513: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. 1 .00 0.80 0.60 Cl' 0.40 0.20 0.00 250.0 350.0 450.0 550.0 (a) 1. 00 0.80 L 2 0.60 C 0.40 0.20 0.00 250.0 350.0 450.0 550.0 Days ( c ) 1 .00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 (b) (d) 550.0350.0 450.0 Days 1.0 0.8 0.6 0.4 0.2 0.0 250.0 350.0 450.0 550.0 Chapter 3. Results^ 37 Figure 3.19: Test 4 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). Days Figure 3.20: Test 4 - RMS interpolated velocity error. Chapter 3. Results^ 38 seems to level off at around twenty percent, but never dropping below the interpolated velocity error. (See Figures 3.23 and 3.24.) The streamfunction contours seemed to compare very well with very little noise, and even the vorticity contours were close. There was a large amount of noise near the boundaries in the vorticity. (See Figures 3.21 and 3.22.) Figure 3.21: Test 5 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) The sixth test case demonstrated a change in the frequency of assimilation to once every ten days. The vorticity error seemed to vary around forty percent and the velocity Chapter 3. Results^ 39 Figure 3.22: Test 5 - Day 513: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. 1 .00 0.80 L 0.60LwT 0.40 cc 0.20 0.00 250.0 350.0 450.0 550.0 (a) 1.00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 (b) 0.20 0.00 250.0 350.0 450.0 550.0 1 .00 0.80 L2 0.60Lw Cl' 0.40mm 1.00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 1 . 0 0.8 0.6 0.4 0.2 0.0 250.0 350.0 450.0 550.0 Days Chapter 3. Results^ 40 Days^ Days ( c ) (d ) Figure 3.23: Test 5 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). (Errors greater than unity truncated at unity.) Figure 3.24: Test 5 - RMS interpolated velocity error. Chapter 3. Results^ 41 around ten percent over the period of the experiment. The assimilated velocity error dropped below the interpolated velocity error on the assimilation steps. The streamfunc- tion error had the recurring oscillation. (See Figures 3.27 and 3.28.) The streamfunction contours had a smooth, similar pattern in the comparisons Figure 3.25(b) and 3.26(b), and the actual values were quite good at day 387, but seemed to be shifted by a constant at day 508. There was, once again a build-up of noise in the vorticity contours on both days. Figure 3.25: Test 6 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) Chapter 3. Results^ 42 Figure 3.26: Test 6 - Day 508: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. 0.00 250.0 350.0 450.0 550.0 0.00 250.0 350.0 450.0 550.0 1.00 0.80 0.60 0.40 0.20 1 .00 0.80 2 0.60 m 0.40 0.20 1.00 0.80 L2 0.60L Cl) w 0.40 0.20 0.00 250 0 350.0 450.0 550.0 Days Days 1 .0 0.8 0.6 0.4 0.2 0.0 250.0 350.0 450.0 550.0 Chapter 3. Results^ 43 (a) ^ (b ) (c ) ^ Id ) Figure 3.27: Test 6 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). Days Figure 3.28: Test 6 - RMS interpolated velocity error. Chapter 3. Results^ 44 The final test case used assimilation every five days, but the data were only available every ten days. Linear interpolation was used to obtain the data values for every second assimilation step. Overall the behaviour seemed to be much like that of test case two. (Compare Figures 3.31 and 3.11.) The RMS streamfunction and vorticity errors seemed to be a touch higher in test case seven. The interpolated velocity error is the same as in experiment six, and the assimilation model velocity error compares well. As in test case two, the contour plots compared well and there was noise evident in the vorticity contours. (See Figures 3.29 and 3.30.) Chapter 3. Results^ 45 Figure 3.29: Test 7 - Day 387: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is no assimilation. (The last assimilation was done at day 383.) Chapter 3. Results^ 46 Figure 3.30: Test 7 - Day 508: A comparison of the (a) reference streamfunction, (b) model streamfunction, (c) reference vorticity, and (d) model vorticity at a time step when there is assimilation. 1.00 0.80 L2 0.60L w 0.40 0.20 1.00 0.80 2 0.60 C" 0.40 0.00 250.0 350.0 450.0 550.0 (a) 1111111A11141111111111/11 0.20 0.00 mittimmigitititlittittitt 250.0 350.0 450.0 550.0 Days (c) 1.00 0.80 0.60 0.40 0.20 0.00 250.0 350.0 450.0 550.0 (b) 1.00 - 0.80 - 0 . 00 ^ 350.0 450.0 550.0 Days (d) 0.60 0.40 0.20 250.0 Chapter 3. Results^ 47 Figure 3.31: Test 7 - RMS Errors: (a) interior velocity, (b) interior vorticity, (c) energy, and (d) streamfunction (including boundary). Chapter 4 Conclusions The simplified adjoint scheme performed very well in the case where there were data available at every interior point on the numerical grid as expected. In the case of having data at every second node, the adjoint scheme showed good results when compared to a standard bicubic interpolation scheme. (See test cases two, four, five, and seven.) There was a need for a smoothing parameter or penalty term added to the cost function to allow the scheme to converge and perform well. There was still noise evident, but could possibly be improved with the choice of the smoothing parameter or perhaps a better smoothing mechanism. An ideal smoothing could not be found to remove the noise and allow the model to converge in test case three with data at every third node, but seemed to be possible in test case five with the larger eddies. An interesting test (test case seven) was the use of linear interpolation in time to increase the frequency of assimilation in time, without requiring observations at each assimilation step. The results compared very well with a similar experiment with data at every assimilation step. (Compare test cases two and seven.) A problem in all cases was the choice of a smoothing parameter. Overall the bicubic interpolation scheme gave a better RMS velocity error, but the observed data were arranged in an ideal pattern for interpolation as well as there being no noise in the data. The interpolation scheme required velocity values at four surrounding points on a grid for each interpolation point as well as the first derivatives (or a finite difference approximation.) Since the data were arranged in a regular grid, the method for obtaining the four closest data values and their derivatives was easy to implement. Given 48 Chapter 4. Conclusions^ 49 a more irregular arrangement of data, the interpolation would be more complicated. Also, the interpolation scheme fits a surface to the data values, giving no error at the points where there are data. If there were any observational errors in the data, they would carry directly through in the interpolation. In most cases the assimilation model RMS error values seem to vary uniformly around a fixed value, except in test case four where the RMS vorticity error dropped significantly. It seemed that the Orlanski radiation condition at the boundaries was not able to prevent RMS error growth between assimilation steps. This problem was intended to be a preliminary investigation of a simple adjoint method. An interesting test would be to compare the assimilation model and the in- terpolation scheme with an irregularly arranged set of observed data and include some random noise. Bibliography [1] Arakawa, A., 1966. Computational design for long term numerical integration of the equations of fluid motion: Two dimensional incompressible flow. Part 1. J. Comput. Phys., 1, 119-143. [2] Bennett, A. F., 1992. Inverse methods in physical oceanography. Cambridge Univer- sity Press. 346 pages. [3] Bryan, K., 1963. A numerical investigation of a nonlinear model of a wind-driven ocean. J. Atmos. Sci., 20, 594-606. [4] Charney, J. G., R. FjOrtoft, and J. von Neumann, 1950. Numerical integration of the barotropic vorticity equation. Tellus, 2, 237-254. [5] Engquist, B. and A. Majda, 1977. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31, 629-651. [6] Gill, A. E., 1982. Atmosphere - Ocean Dynamics. Academic Press, Inc., New York. 662 pages. [7] LeBlond, P. H. and L. A. Mysak, 1978. Waves in the Ocean. Elsevier Science Pub- lishers B. V., Amsterdam, The Netherlands. 602 pages. [8] McWilliams, J. C. and G. R. Flierl, 1976. Optimal, quasi-geostrophic wave analyses of MODE array data. Deep -Sea Res., 23, 285-300. [9] Miller, R. N. and A. F. Bennett, 1988. Numerical simulation of flows with locally characteristic boundaries. Tellus, 40A, 303-323. [10] The Numerical Algorithms Group Limited, 1991. The NAG Fortran Library Manual, Mark 15. [11] Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys., 21, 251-269. [12] Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in Fortran: The Art of Scientific Computing, Second Edition, Cambridge University Press. 963 pages. [13] Raymond, W. H. and H. L. Kuo, 1984. A radiation boundary condition for multi- dimensional flows. Quart. J. R. Met. Soc., 110, 535-551. 50 Bibliography^ 51 [14] Robinson, A. R. and D. B. Haidvogel, 1980. Dynamical forecast experiments with a barotropic open ocean model. J. Phys. Oceanogr., 10, 1909-1928. [15] Thacker, W. C., 1987. Three lectures on fitting numerical models to observations. External report GKSS 87/E/65., GKSS-Forschungszentrum Geesthacht GmbH, Geesthacht, Federal Republic of Germany, 64 pages. [16] Thacker, W. C. and R. B. Long, 1988. Fitting dynamics to data. J. Geophys. Res., 93, 1227-1240. [17] Tziperman, E. and W. C. Thacker, 1989. An optimal-control / adjoint-equations approach to studying the oceanic general circulation. J. Phys. Oceanogr., 19, 1471- 1485. [18] Veronis, G., 1966. Wind-driven ocean circulation - Part 1. Linear theory and per- turbation analysis. Deep-Sea Res., 13, 17-29. [19] Veronis, G., 1966. Wind-driven ocean circulation - Part 2. Numerical solutions of the non-linear problem. Deep-Sea Res., 13, 31-55. [20] Zou, J. and W. W. Hsieh, 1993. Open boundary control in a limited-area ocean model through data assimilation. (in progress)
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