Observations and Modelling of Fronts and Frontogenesis

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APPENDIX B: NUMERICAL METHOD The equations solved numerically were (28), (30), and (33), with the boundary conditions (17) and the matching conditions (20),(23), and (29) when appropriate, along with the evaluations (32) and (34). The initial data (27) were given on a grid of 150 points, logarithmically spaced for small y and linearly spaced for large y. Upper layer characteristics were followed continuously, with new points added both adjacent to the boundary and in the interior when divergence in the mixed layer separated grid points. De Szoeke and Richman (1984) successfully integrated the interior layer potential vorticity "backward" (with respect to the interior layer characteristic curves) along the mixed layer characteristic curves. It was necessary to maintain accuracy over longer time scales in the present study, so this approach was abandoned in favor of integrating the interior layer variables along their appropriate characteristic curves. Because grid points moved at different speeds and in different directions along the characteristic curves in the different layers, the values of the interior layer variables were interpolated onto the layer 1 grid for solution of the diagnostic equations (28), and the resulting interior velocities v2 and v3 interpolated back onto the interior layer grids for time-stepping. A local quadratic interpolation proved sufficient to maintain
smoothness. The large vorticities that arose in the boundary layer at the coast made spline interpolation impractical. The interior layer grids, which converged toward the coast, were reset regularly, each time the third interior point passed inshore of the second layer 1 point (the first points in each layer were fixed to the coast). The boundary value problem (28) was solved by inverting the five-diagonal matrix finite difference form of (28). Two extra upper diagonals were included to ensure numerical stability in the implementation of the regularity condition (29) at the juncture of the two- and three-layer subdomains. We were unable to obtain a stable numerical estimate of dy2/dt for integration of the evolution equation (19) for the location of the juncture between the two- and three-layer subdomains. This was apparently due to the effect of numerical noise, which became appreciable near the point y because of the small layer depths h2. Consequently, the location y had to be determined implicitly at each time. This was achieved in two steps. First, h2 was required to vanish on any layer 2 characteristic curve on which it had vanished at any previous time. Second, because by (11) and (13) the layer 3 potential vorticity P3 was never less than the value it retained as long as layer 3 remained isolated from the mixed layer, the layer depth h3 calculated from the known geostrophic vorticity and conservation of potential
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AN ABSTRACT OF THE THESIS OF Roger
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Observations and Modelling of Front
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To my parents, Hans and Nancy Samel
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This research was supported by ONR
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TABLE OF CONTENTS I. INTRODUCTION 1
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Figure LIST OF FIGURES (continued)
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OBSERVATIONS AND MODELLING OF FRONT
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II. TOWED THERMISTOR CHAIN OBSERVAT
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oriented multiple surface fronts up
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gravity waves. A maximum of 21 and
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small, since the large-scale temper
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front at about 18.5 °C. There is f
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given in the stable cases (downward
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temperatures for January and Februa
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dominates the spectrum. At these wa
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typically iO-2iO3 °c m1-), and sin
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11.5 Geostrophic turbulence We have
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energy of each of the longitudinal
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varying N and a flow isolated from
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adiabatic meridional displacements
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z0 34 32 V v 30 a -J 28 26 a) - 2b$
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U0 V L 0 L V 20 E 16 V 12 p ; t *,
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30 E 50 -c a- I, 0 70 90 300 360 37
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10 10° 10-2 1 0 -3.0 -2.0 -1.0 0.0
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1 o 1 102 101 - 0 LU 00 x I.- - -1
Page 51 and 52: E (I) a. z 106 1 o5 102 101 100 1O
Page 53 and 54: Table 11.1 Tow start and finish pos
Page 55 and 56: Table 11.3 Climatological sea surfa
Page 57 and 58: Paulson, C. A., R. J. Baumann, L. M
Page 59 and 60: 111.1 Introduction In this chapter
Page 61 and 62: interior layers, on the horizontal
Page 63 and 64: polynomials in z.) The vertical mom
Page 65 and 66: hit + (hivi) = We, (8a) h2t + (h2v2
Page 67 and 68: Equations (la) and (2a) are dynamic
Page 69 and 70: subdivided into subdomains with dif
Page 71 and 72: mixed layer. As is the case with (2
Page 73 and 74: constant pressure c = 4.1x103 J K1-
Page 75 and 76: When (28) have been solved for the
Page 77 and 78: occur with the initial conditions w
Page 79 and 80: analytically. Let Al2 and A23 be th
Page 81 and 82: Shortly after t = 8.1, the mixed la
Page 83 and 84: that around the offshore front, exc
Page 85 and 86: the reduced equations numerically.
Page 87 and 88: (1) The upwelled horizontal profile
Page 89 and 90: Figure 111.1 Model geometry. zO z z
Page 91 and 92: * j 0. -20. >\ - -40. 0 > .0 0. 40.
Page 93 and 94: 4) 20. 10. 0. 20. 1 0. 0. 0 y/X 0.
Page 95 and 96: BIBLIOGRAPHY Batchelor, G. K. , 195
Page 97 and 98: Rhines, P. B., 1979: Geostrophic Tu
Page 99 and 100: APPENDIX A: DYNAMICAL DERIVATION OF
Page 101: In a manner exactly analogous to th
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