Climate System Modeling Notes PDF - University of Bologna

University of Bologna Department of Physics and Astronomy Master Degree in Science of Climate ∼·∼ Academic Year 2024–2025 Climate System Modeling NOTES Prof....

University of Bologna Department of Physics and Astronomy Master Degree in Science of Climate ∼·∼ Academic Year 2024–2025 Climate System Modeling NOTES Prof. Student Prof. Antonio Navarra beautiful babies 00000 Date: December 11, 2024, download latest 2 The contents of these notes stem from the course of Climate System Modeling taught by Prof. Antonio Navarra1 at University of Bologna for the academic year 2024/2025 and from his notes available at wanderer.cmcc.it2. The source code is available on Github. 1 CMCC, Unibo 2 If prompted, use the credentials: { user: cmcc , psw: pD8phJg3e76J } Contents 1 Physical description of the climate system 7 1.1 Atmosphere..................................... 7 1.1.1 The General Circulation of the Atmosphere............... 8 1.1.2 The time averaged zonal general circulation............... 10 1.1.3 The horizontal general circulation wind................. 10 1.1.4 Mean Sea Level Pressure and The Sea Surface Temperature...... 11 1.2 Oceans........................................ 13 1.2.1 The salinity structure of the ocean.................... 14 1.2.2 Influence of the ocean........................... 17 1.3 Ice.......................................... 21 1.4 Interconnections between atmosphere, ocean and ice.............. 21 1.4.1 Atmosphere effect............................. 22 1.4.2 Ocean effect................................. 22 1.4.3 Ice...................................... 23 2 Fundamental Equations and processes 25 2.1 Fundamental Equations.............................. 25 2.1.1 Introduction................................. 25 2.1.2 Coordinate systems............................. 25 2.1.3 Equation of motion............................. 27 2.1.4 Summary of fundamental equations-................... 31 2.1.5 Simplified equations-............................ 31 2.1.6 Linear solutions............................... 34 2.1.7 Waves.................................... 34 2.2 Homogeneous flows................................. 35 2.2.1 The vorticity equation on the β-plane.................. 38 2.3 Rossby waves.................................... 39 2.4 Fundamental processes............................... 40 2.4.1 Radiation.................................. 40 2.4.2 Moisture................................... 44 2.4.3 Clouds.................................... 46 2.4.4 Surface Processes.............................. 47 2.4.5 Hydrology.................................. 48 3 Numerical methods used in atmospheric models 49 3.1 Historical introduction............................... 49 3.2 Methods for the numerical solution of the equations of motion......... 50 3.3 Grid point method................................. 51 3.3.1 Finite difference schemes.......................... 53 3.3.2 Convergence................................. 55 3.3.3 Stability................................... 56 3.4 Time differencing schemes............................. 60 3 4 CONTENTS 3.4.1 Properties of schemes applied to the oscillation equation........ 63 3.4.2 A combination of schemes......................... 68 3.5 The Advection Equation.............................. 68 3.6 Energy and numerical methods.......................... 72 3.6.1 Conservation in numerical schemes.................... 74 3.6.2 The Arakawa Jacobian........................... 75 3.6.3 Gravity waves................................ 77 3.6.4 Two dimension............................... 79 3.7 Spectral Methods.................................. 83 3.7.1 Numerical spectral solution (nonlinear).................. 86 3.8 Models on the sphere/Spherical frame...................... 87 3.8.1 The Helmholtz theorem.......................... 89 3.8.2 Spherical Harmonics............................ 90 3.8.3 Spectral Transforms............................ 91 3.8.4 Truncation................................. 94 3.8.5 Summary.................................. 94 3.9 Lagrangian Methods................................ 98 3.9.1 Semi-Lagrangian Methods......................... 99 3.10 Vertical coordinates................................ 100 4 Climate Models 103 4.1 Radiative Model.................................. 103 4.1.1 Two-layer atmosphere........................... 106 4.1.2 Three-layer atmosphere.......................... 107 4.1.3 Radiative convective equilibrium..................... 109 4.2 Radiative forcing.................................. 111 4.2.1 Greenhouse feedback............................ 113 4.2.2 Water vapor feedback........................... 114 4.3 Radiative-Convective Model............................ 116 5 Models 119 5.1 Ocean models.................................... 119 5.2 Ice models...................................... 122 5.3 The River Transport................................ 122 5.4 Grids......................................... 122 5.4.1 Regular grids................................ 122 5.4.2 Unstructured grids............................. 123 5.4.3 Adaptive Grids............................... 123 5.4.4 Vertical Grids................................ 124 5.4.5 Resolution and mesoscale eddies..................... 125 5.5 Caos......................................... 128 5.6 Vorticity dynamics................................. 128 5.6.1 What maintains the circulation?..................... 128 6 Exploring Variability 131 6.1 Designing a model for variability......................... 134 6.2 General circulation model............................. 136 7 SST Forcing 141 7.1 How can surface sea temperature impact atmospheric circulation?....... 142 7.1.1 What is the mechanism by which SST is affecting climate?....... 145 7.1.2 Steady heating balances.......................... 146 CONTENTS 5 8 Earth System Models 153 8.1 Expectations with increasing greenhouse gases.................. 154 8.2 Expanding the complexity of climate modeling................. 155 8.3 Carbon and Biogeochemical cycles........................ 156 8.3.1 The Carbon cycle.............................. 157 8.3.2 Complexity and model evaluation..................... 159 8.4 Digital Twins.................................... 162 8.4.1 Enabling Technologies........................... 162 8.5 Climate Ensembles................................. 163 9 Theory of symmetric circulation 165 10 Theory of symmetric circulation 167 11 Tipping points 173 11.1 Examples of Climate System Tipping Points................... 174 11.2 Climate impacts.................................. 176 12 Are General Circulation models obsolete? 179 A Mathematical Complements 181 A.1 Vector calculus................................... 184 A.1.1 Curl..................................... 184 A.1.2 Gradient................................... 184 A.1.3 Divergence................................. 184 A.2 EOF: Empirical Orthogonal Functions...................... 185 A.2.1 Eigenvalue decomposition → Singular Value Decomposition...... 186 A.2.2 Limitations................................. 187 A.2.3 Variance................................... 187 A.2.4 Covariance................................. 187 A.2.5 Correlation................................. 187 A.3 Space-time splitting................................ 187 A.3.1 Zonal mean................................. 187 A.3.2 Time mean................................. 188 A.3.3 Higher order quantities........................... 188 Bibliography 191 6 CONTENTS Chapter 1 Physical description of the climate system Last updated: 2024-12-05 Source file: chapter-intro.tex 1.1 Atmosphere Composition The atmosphere is a gaseous envelope that surrounds the Earth, acting as a protective envelope essential for life. It is composed of a mechanical mixture of several gases, with nitrogen being the most abundant (78% of the volume of dry air), followed by oxygen (20.95%), argon (0.93%), and carbon dioxide (0.037%). Trace gases like neon, helium, methane, and hydrogen also contribute to its composition, along with varying amounts of water vapor, which depend on factors such as location, evaporation rates, and temperature. This gaseous mixture is held to the Earth by gravity, which causes the atmosphere to exert pressure. Near the surface, the pressure is greatest, approximately 1.013 × 103 mb at sea level, and it decreases with altitude. This reduction in pressure reflects the compressibility of gases, making the atmosphere densest near the ground. This characteristic distinguishes the atmosphere from the ocean, where the density remains relatively uniform due to the near incompressibility of liquids. The Vertical Temperature Gradient The atmosphere exhibits a distinct temperature profile that changes with altitude and varies depending on the region and season. These variations define the layers of the atmosphere, each with unique characteristics and processes. The Troposphere: is the lowest layer, extending from the surface up to about 17 kilo- meters in the tropics and 8–9 kilometers in polar regions. In this layer, the temperature generally decreases with altitude at an average rate of 6.4◦ C km−1. This gradient arises because the surface absorbs heat from the sun and transfers it to the air above, while higher altitudes are farther from this heat source. The troposphere is also the region where most weather phenomena, including clouds and storms, occur, making it the most dynamic part of the atmosphere. At the upper boundary of the troposphere, known as the tropopause, the temperature reaches its minimum and remains constant for a short distance. In tropical regions, the tropopause is particularly cold, with temperatures dropping to about 190 K, marking one of the coldest points in the Earth’s atmosphere. 7 8 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM The Stratosphere: Above the tropopause lies the stratosphere, a more stable layer where temperatures either remain constant or begin to rise with altitude. This increase is primarily due to the absorption of ultraviolet radiation by ozone molecules, which warms this part of the atmosphere. Unlike the troposphere, the stratosphere lacks the turbulence and weather systems associated with surface heating. The Thermosphere: the temperature continues to increase steadily with altitude. This layer is less influenced by surface-level processes and is instead heated by the absorption of high-energy solar radiation, making it the hottest layer of the atmosphere. Troposphere (sum/win) Tropopause Stratosphere poles 280 / 235 9 km: 230 / inversion (1.5/2 km) Slow Warming/ slow cooling midlat 290 / 280 10 km: 220 Slow Warming/ slow cooling tropics 300 17 km: 190 warming Regional and Seasonal Variations Temperature profiles in the atmosphere are not uniform and vary significantly between regions and seasons. Polar regions: In summer, temperatures near the surface reach about 280 K, decreas- ing gradually with height. In winter, surface temperatures can plummet to 235 K, but a slight increase often occurs at altitudes of 1.5–2 kilometers due to a phenomenon called temperature inversion. This inversion, caused by the cooling of surface air, reverses the usual trend of decreasing temperature with height. Tropical regions: Surface temperatures remain relatively constant, averaging around 300 K. However, as altitude increases, the temperature drops sharply, reaching the coldest point at the tropopause (about 190 K). Mid-latitudes: The temperature profile combines characteristics of both polar and tropical regions, varying more noticeably between seasons. 1.1.1 The General Circulation of the Atmosphere Since heated air tends to rise and low-level air must flow in to replace the risen heated air, the relative heating and cooling of different areas in the Earth’s surface plays a significant role in driving local winds and the large-scale atmospheric circulation. The atmospheric circulation can be summed up into two different contributes: the east-west motion of the Walker Circulation and the north-south circulation of the three-cell structure: Hadley, Ferrel and polar cells. The Coriolis effect plays a crucial role in wind deflection to the right in the Northern Hemisphere and to the left in the Southern Hemisphere explaining the formation of trade winds, westerlies, and polar easterlies. 1. - The Walker circulation refers to the equatorial Pacific and involves rising air over the region of Indonesia and descending air over the eastern Pacific. It was examined following the discovery of strong negative correlation between surface pressure anomalies in the two regions. Variations in the strength of this circulation produce a large scale fluctuation with an irregular period, known as Southern Oscillation. 2. Three-cell pattern The tropical Hadley cell is driven by solar heating, causing rising motion near the Equa- tor, then by the release of latent heat as the rising air leads to precipitation in the 1.1. ATMOSPHERE 9 rising-air branch of the cell. This rising-air branch of the Hadley cell is not centered consistently on the equator but migrates north & south. Moreover, the strength of the Hadley circulation varies with longitude, being strongly affected by such factors as whether the underlying surface is land or ocean. After rising near the Equator, the air in the Hadley cell moves poleward, sinking near 30°N and 30°S and thereby generating belts of surface-level high pressure near 30°N and 30°S. Since the sea level pressure is low, the high pressure produced by the sinking air at about 30°N and 30°S creates a surface pressure gradient leading to the movement of a portion of the sinking air back to- ward the low pressure near the equator. The region of low-level convergence toward the bottom of the rising-air branch of the Hadley cell is called the intertropical convergence zone (ITCZ). The position of the ITCZ shifts with the seasons, moving north during the Northern Hemisphere’s summer and south during its winter. This seasonal movement affects global wind and rainfall patterns, contributing to phenomena like monsoons. In the mid-latitude Ferrel cell, termed indirectly because of having rising air in its cooler branch, the low-level flow is toward the poles, away from the relatively high pressure produced by the descending arms of the Hadley and Ferrel cells at about 30° latitude and toward the relatively low pressure at about 60° latitude. The polar cell : strong radiational cooling near the poles, causes polar air to become cold and dense, which in turn causes it to sink. Thus there is relatively high pressure at the pole, which, combined with the low pressure near 60°N and 60°S discussed in connection with the rising-air branch of the Ferrel cell, produces surface flow equatorward from the pole. This polar cell is extremely weak, although it remains detectable in time averages of the air circulation. Figure 1.1: Global Atmospheric Circulation cells In reality, the atmospheric circulation is more complex than the three-cell model suggests. It exhibits significant temporal and spatial variations due to factors like land/sea contrasts, to- pography, and changes in solar heating. While the model serves as a foundational framework, modern climate science relies on satellite data and simulations for a detailed understanding of global atmospheric dynamics. Monsoons and Land Sea breezes: they are wind patterns driven by differences in heating between land and water surfaces, but they operate on different scales. Sea Breeze (Daytime): During the day, land heats up faster than water, causing air over the land to rise and creating a low-pressure zone. Cooler, denser air from over the sea flows in to replace it, generating a breeze from sea to land. Land Breeze(Nighttime): At night, the land cools faster than water, creating a high-pressure 10 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM zone over the land. Air flows from the land to the sea, forming a weaker land breeze. Summer Monsoon: Similar to a large-scale sea breeze, during summer, land masses (like South Asia) heat up more quickly than the surrounding oceans. This creates a low-pressure zone over land, drawing in moist air from the Indian or Pacific Oceans. The moisture-laden air rises and cools, resulting in heavy rainfall, particularly over mountain ranges like the Himalayas and Ghats. Winter Monsoon: In winter, the land cools faster than the ocean, forming a high-pressure zone. Cold, dry air flows outward from the land to the sea, leading to dry weather conditions over most of South Asia. Key Differences:while land/sea breezes are localized and operate on a daily cycle, monsoons occur over larger areas and are seasonal, driven by shifts in the Intertropical Convergence Zone (ITCZ). Monsoons involve additional complexities, including the influence of upper atmospheric circulations, such as the jet stream. The Asian monsoon, particularly over South Asia, is the most prominent example, crucial for regional agriculture and ecosystems. However, its dynamics are far more complex than those of simple land and sea breezes. 1.1.2 The time averaged zonal general circulation The time-averaged circulation, computed from ERA5 Reanalysis data, shows westerly mid- atmosphere jets in the subtropical regions of both hemispheres. These jets reach a maximum around the 200mb level (1̃2 km) and extend to the surface in the mid-latitudes, while easterly flows dominate the equatorial zone. The jets exhibit a strong seasonal cycle, with accelerated jets in winter and weaker jets in summer. There is a seasonal poleward migration of the jet cores, with the winter season showing more concentrated and intense maximums. The Southern Hemisphere’s winter jet is broader than its Northern Hemisphere counterpart, both linked to the stratospheric flow above. The stratosphere also exhibits strong jets with a stronger seasonal cycle, where easterlies replace westerlies as the seasons change. In the meridional circulation, low-level convergence occurs at the equator, with high-level divergence. The InterTropical Convergence Zone (ITCZ) oscillates between 15°N and 5°S, influenced by the seasonal cycle of the sun. Temperature decreases with latitude due to radiation balance, with the equator receiving more solar radiation than the poles. The temperature decreases with height in the atmosphere, though the lapse rate is less than adiabatic, indicating a generally stable atmosphere. There is a strong latitudinal temperature gradient, which weakens with altitude and reverses in the upper atmosphere and stratosphere, due to radiation absorption by ozone and other components in the lower stratosphere. The reversal of the meridional temperature gradient aligns with the zonal wind structure, showing positive shear in the troposphere and negative shear above, consistent with ther- mal wind balance. Water vapor, measured as specific humidity, is concentrated in the lower atmosphere and the equatorial region, with significant dryness at higher altitudes. The at- mosphere’s moisture content decreases with altitude due to the Clausius-Clapeyron relation, which links water vapor pressure to temperature. At the equator, moist air contains 12-15 grams of water per kilogram of air. Specific humidity follows the seasonal cycle of the sun, shifting latitudinally. The max- imum is located around 5°S in December-February (DJF) and around 10°N in June-July- August (JJA). The winter hemisphere is more moist at the surface than the summer, though values in the winter are smaller than in the equatorial zone, reaching up to 8 g/kg. ] 1.1.3 The horizontal general circulation wind The climatological geopotential height at 200mb shows deviations from a zonally sym- 1.1. ATMOSPHERE 11 metric circulation, particularly over the east coasts of continents, downstream from major mountain ranges like the Rockies and Himalayas. These features are clearer with special- ized projections, highlighting the winter atmospheric pattern. While geopotential height can approximate wind flow in mid-latitudes, wind analysis is more useful in low latitudes. At 200mb, winter jet streams are strong in both hemispheres, especially over Asia, with the Asian jet reaching velocities over 70 m/s. The Southern Hemisphere jet is weaker. In summer, westerly jets are weaker, and easterly jets dominate the tropics. The Indian Ocean sees a strong easterly flow in summer, leading to divergence over South America and Indonesia. The meridional wind shows alternating poleward and equatorward patterns in winter, with large deviations in the Pacific-North American sector and East Asia. In summer, winds weaken but remain stronger in the Southern Hemisphere. Near the surface, the zonal wind is stronger over oceans and weaker over land, with easterlies in the subtropics, except in the Indonesian region, where westerlies prevail. The equatorial Pacific shows a convergence zone. During JJA, the seasonal cycle strengthens winter wind features, with strong westerlies in the Indian Ocean signaling the start of the South Asian Monsoon. The meridional wind shows an equatorward flow along continent coasts, intensifying in summer, and reversing along the Somali coast. At the 850mb level, trade winds dominate the equatorial Pacific, shifting with the ITCZ’s seasonal cycle. The Asian Summer Monsoon is visible in the Indian Ocean, forming a large gyre from East Africa to the Indian subcontinent, crucial for understanding low-level atmo- spheric and oceanic interactions. 1.1.4 Mean Sea Level Pressure and The Sea Surface Temperature (MSLP) is a key parameter in describing atmospheric circulation. It represents the atmo- spheric pressure adjusted to mean sea level, though its usefulness is limited in areas with significant mountains, like the Rockies, Himalayas, and Antarctica, where the concept of "sea level" is effectively underground. Outside these areas, MSLP provides a good representation of atmospheric mass distribution. High MSLP areas correspond to mass accumulation, par- ticularly in the subtropics, where high-pressure systems are common in both summer and winter. Near-surface temperatures, typically measured 2 meters above the ground, reflect the seasonal and geographical variation in temperature. Over oceans, they follow sea surface temperature (SST), but land areas show significant seasonal changes. Northern continents are colder than the oceans at the same latitude, and coastal regions also experience temperature differences, with west coasts being milder than east coasts. In winter, high pressure tends to remain over land while low pressure develops over the oceans, reversing in summer. The intertropical zone sees high-pressure centers shifting with the seasons, while the Southern Hemisphere features a more symmetric pattern, with a ring of low pressure around Antarctica. The seasonal cycle is evident in the shift of high-pressure areas in the tropics, moving latitudinally with the changing seasons. The meridional distribution of sea level pressure shows high pressure in the tropics, with low pressure areas near the equator and in the mid-latitudes. In the Southern Hemisphere, the pressure gradients are stronger and more pronounced, with tropical pressure maxima shifting seasonally. Sea Surface Temperature (SST) shows strong north-south gradients, with polar regions being cold and the equator generally warm. However, notable deviations from zonal symmetry occur near continental east coasts, particularly in the equatorial Pacific, where cold water intrudes at the equator in the East Pacific, contrasting with the warm waters of the West Pacific. These temperature differences highlight significant gradients in SST along the equator 12 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM The Role of Energy Processes The vertical temperature gradient in the atmosphere is maintained by complex energy pro- cesses, including the absorption and radiation of heat, convection, and interactions with the Earth’s surface. These processes vary with latitude and season, creating the dynamic and layered structure of the atmosphere. Inversions, which occur when surface air cools significantly, disrupt the usual temperature gradient, particularly in the troposphere. These events highlight the intricate balance of energy transfer within the atmosphere and its impact on weather and climate. Energy balances To understand how the atmosphere maintains the various temperature structures, it is neces- sary to consider the processes by which the atmosphere gains and loses energy and how these processes vary with latitude and season. Essentially all the energy that enters in the climate system comes from the sun, part of it is absorbed in the system and must be balanced with outgoing energy leaving the system, to maintain the overall climate system in its observed equilibrium state. Because of the different temperatures between the sun and the Earth, the energy emitted by the two bodies is sharply different. The spatial and temporal distribution of the receipt of solar energy at the Earth’s surface is highly dependent upon the Earth’s annual revolution around the sun, its daily rotation about its axis, and the tilt of its axis concerning the plane of its orbit. When the Earth is in the portion of its orbit where the Northern Hemisphere is tilted toward the sun, the Northern Hemisphere receives the majority of the direct sunlight and ex- periences summer, while the Southern Hemisphere experiences winter. Naturally, the reverse occurs when the Earth is in the opposite portion of its orbit, with the Southern Hemisphere tilted toward the sun. About half of the incoming solar radiation is absorbed by the Earth’s surface. This energy is transferred to the atmosphere by warming the air in contact with the surface (sensible heat), by evaporation and by thermal radiation that is absorbed by clouds and by greenhouse gases. The atmosphere in turn radiates thermal radiation back to Earth as well as out to space. Since all the incoming radiation equals to 342 Wm−2 and 107 Wm−2 of it is reflected back to space, there must be 235 Wm−2 of terrestrial radiation emitted to space for an equilibrium situation from three main sources: water vapor, CO2 , clouds and Earth’s surface. At the top of the atmosphere, the energy balance is composed of the compensating fluxes between the incoming solar radiation and the outgoing thermal radiation from the Earth. The solar radiation will be modulated by the reflectivity caused by the atmospheric cloud and in smaller part by the molecular components of the atmosphere itself. The thermal radiation will be modulated by the absorption properties of the atmosphere and its constituents. Both will be affected by the circulation and ultimately by the atmospheric flows. The net balance shows a surplus of radiative flux in the subtropical region and a deficit in the polar regions. The global radiation budget can be obtained by integrating over the surface of the Earth the radiative fluxes. At the surface, the energy balance involves more processes. There is also here a net solar radiation flux, modulated by the albedo of the surface and of the atmosphere, and a net thermal flux, obtained as the balance between the radiation emitted by the surface and the downward flux coming from the bulk of the atmosphere. Then there is a sensible heat flux caused by the turbulent vertical motion that carries away heat through mechanical agitation and the latent heat flux that represents the heat necessary for the evaporation of water from the surface. Because of the extent of the ocean, the latent heat flux is an important element of the budget. 1.2. OCEANS 13 In addition, we can look at the total cloud cover which is essentially the aggregated quantity of clouds at various levels. This is the quantity that is intuitively linked to the observation of a “cloudy sky”. In general, clouds are present over a vast surface of the globe. Some areas show a persistent absence of cloudiness over the entire year, such as the Sahara desert, part of the Arabian peninsula, Australia, and South Africa. Other areas instead show a strong seasonal cycle, with different could cover in different seasons. It is interesting to note that the seasonal cycle takes different characteristics in different areas. In the mid-latitudes cloud cover is larger in the local hemispheric winter than in the summer. Looking at the zonal profile of the clouds, it is evident that major concentrations are related to the equator and middle/high latitudes. 1.2 Oceans Oceans, covering approximately 71% of Earth’s surface, play a crucial role in global climate systems by storing and transferring heat, nutrients, and momentum. The interaction between the atmosphere and oceans significantly impacts weather and climate patterns. These bodies of water contain an estimated 1,350 million cubic kilometers of water, predominantly found in the Pacific, Atlantic, Arctic, and Southern Oceans, with an average depth of about 4,000 meters. Surrounding continents are continental shelves, shallow areas that extend outward for several kilometers before dropping sharply at the continental slope into the deep ocean floor. The ocean floor itself is varied, featuring features like underwater mountains, ridges, trenches, and smooth sedimentary basins. The oceans redistribute solar energy from equatorial to higher latitudes. As water warms at low latitudes, it absorbs significant amounts of solar heat. This heat is released into the atmosphere in the form of latent heat and longwave radiation as water cools or condenses, influencing global atmospheric circulation. Ocean currents further distribute this heat, bal- ancing regional temperatures and aiding in climate regulation.(*) Composition. Seawater is not just water but a mixture containing various dissolved salts, with an average salinity of 35 grams per kilogram of water. This salinity largely comes from chloride, sodium, and sulfate, accounting for the majority of dissolved material. Salinity levels, however, vary globally. The Arctic Ocean shows lower salinity ( 29%), influenced by melting ice and freshwater inflows, while the subtropical Atlantic has some of the highest salinities ( 37.5%), due to high evaporation and low precipitation. These variations affect density, which in turn drives ocean circulation and temperature dynamics. Overall, oceans are not only a storage medium for heat and nutrients but also critical in redistributing energy across the planet, moderating climate, and supporting ecosystems. Their interaction with atmospheric processes underscores their integral role in Earth’s environmental systems. Temperatures profiles. Surface Temperatures: Range: -1°C (polar regions) to 20–30°C (tropics). Seasonality: temperatures align in east-west zonal patterns, with exceptions like: Tropical Pacific: western regions warmer than eastern regions due to currents. Gulf Stream: transports warm water northeast, moderating European climates. Temperature with Depth: Mixed Layer (0˘30 m in summer): warmed by sunlight and mixed by winds. 14 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM Thermocline (200˘1000 m): sharp temperature gradient separating surface water from deeper layers. Deep Ocean (> 1000 m): uniform cold temperatures (0.5˘1.25C). Coldest water (−0.25C) near Antarctica due to sinking dense, salty water from surface cooling and sea ice formation. R0.5 Ocean circulation is driven by temperature and salinity differences, influencing density and pressure. Surface currents are wind-driven, forming large gyres such as those in the Pacific and Atlantic Oceans. These gyres redistribute heat and nutrients, shaping regional climates and ecosystems. Deep ocean currents, part of the thermohaline circulation1 , involve the sinking of dense, cold water and the upwelling of warmer, nutrient-rich water, critical for sustaining marine life. Additionally, mesoscale eddies, swirling water masses, play a signifi- cant role in transporting momentum, heat, and nutrients horizontally and vertically, although their dynamics are not fully understood. These processes together illustrate the complex and vital role oceans play in regulating Earth’s climate and supporting life. Figure 1.2: The major water masses of the Atlantic Ocean 1.2.1 The salinity structure of the ocean The salinity structure of the oceans is shown in Figure 1.2.1. The top panel shows the salinity near the surface at a nominal depth of 5m. We notice that there is a complex structure with low salinity (fresher) waters at the poles and progressively more saline water moving towards the Equator, but then salinity decreases again at the equator. It is probably instructive to look at the latitudinal distribution of the precipitation. We can notice that the peaks of precipitation in the midlatitude and at the Equator are correlated with the low-salinity areas, whereas the subtropical regions are regions of strong evaporation, whose signature is the high salinity of the surface waters. 1 https://rwu.pressbooks.pub/webboceanography/chapter/9-8-thermohaline-circulation/ 1.2. OCEANS 15 The Pacific Ocean is fresher than the Atlantic or even the Indian Ocean. We can see from the map at 1000m that sources of saline waters are marginal seas like the Mediterranean and the Persian Gulf. These areas are evaporative basins that produce water so saline that it dominates over the temperature effect and becomes denser so that we can find Mediterranean water below the surface. R0.5 The effect of the precipitation is less visible at the deeper depth pf 1000m (bottom panel), where the ocean tends to be fresher and more uniform. We are using here the same scale to give a feeling of the changes in salinity, except for the Atlantic and the east Indian Ocean, the salinity is around 34 psu. The effect of the runoff from major river systems is visible along the Atlantic coast of South America where the Amazon river discharges and in the Bay of Bengal and around Indochina from the run-off the major rivers there, the Ganges and the Mekong. In the pictures below: on the left salinity deviations with depth, going even deeper we have to change drastically the scale. The oceans are remarkably uniform and the salinity deviations are really small. On the right, Pacific and Atlantic ocean salinity. Looking at the North Atlantic (figures below) we notice that there is a strong salinity gradient along the North American Coast that follows roughly the pattern of the temperature gradient shown below. Strong temperature gradients are presumably to be connected to the existence of currents, but we will need to check the density, depending on the salinity later, to be sure. Anyway, this picture is giving a strong indication of the existence of something remarkable 16 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM and intense along the western boundary of the Atlantic ocean. A similar situation exist in the North Pacific along the Japan coast and therefore we can start to suspect that this has to do with the presence of the continental boundary. The previous analysis of the temperature is giving us hints of a strong vertical structure of the oceans, so it may be useful to look at the vertical distribution somewhat more in detail. The figures above show the same section North-South section of along the longitude of 25W, roughly in the middle of the Atlantic Ocean. The salinity follows a similar pattern as the temperature with a strong gradient approximately at the thermocline, the high salinity is confined in thin; layers at the surface, but some interesting behaviour is visible below. In the Northern Hemisphere we can see the Mediterranean water penetrates at depth and actually protruding under the fresher water of Antarctic origin that is colder, but because is fresher floats over the Mediterranean water. Really cold Antarctic water reaches the bottom, filling the abyssal plains of the basin. Figure 1.3: North-South Salinity section at 25W longitude. The Mediterranean waters are clearly visible in a longitude-depth section (Figure 1.3). The saline mediterranean water sinks to about 1000m because it is warmer than the Atlantic but much more saline so it is denser and it reaches an equilibrium depth at about 1000m. Figure 1.4: The vertical structure of the Pacific Ocean The vertical structure of the Pacific Ocean (Figure 1.4) is different and what we see is a situation where salinity is slowly varying getting fresher toward the surface, with the thin saline water in the subtropics a clear signature of the equatorial upwelling and the Antarctic water filling the abyssal plains. The Indian Ocean is similar to the Pacific South of the Equator but North of the Equator at this longitude is fresher. The Indian Ocean (Figure 1.5) shown here as a section cutting essentially through the Bay of Bengal, shows a similar strong stratification, but there are only weak signs of an equatorial upwelling of cold water. We can gain more insights in the equatorial structure by looking at a longitudinal section along the Equator in the Pacific (Figure ??). Here we see the general difference between the Atlantic and the Pacific, but we also see that the salinity is following the slant of the 1.2. OCEANS 17 Figure 1.5: As in Fig. 1.4 but for 90E longitude equatorial thermocline and the equatorial upwelling in the East Pacific. The effect of the major precipitation center of ITCZ in the West Pacific is visible in fresh water at the surface in the West. 1.2.2 Influence of the ocean The continentality effect It makes continental summers warmer and winters cooler than the adjacent ocean areas. The oceans have this effect because of their large heat storage capacity and because part of the energy that they retain from solar input during the summer they return to the atmosphere in the forms of sensible and latent heat and longwave radiation the following winter. Conversely, the continents have the reverse effect, strengthening the seasonal temperature contrasts Ocean currents They can transport heat from low to high latitudes. (in the Gulf Stream of the North Atlantic). This contributes to much higher winter air temperatures over mid-to- high latitude oceans than over adjacent land areas at the same latitude. The overall basin circulation The surface currents of the Atlantic ocean are shown in Figure 1.6. As we might have suspected from the temperature coast that reaches all the way across the North Atlantic to Europe, the Gulf Stream. Strong currents are also visible in the Equatorial area where they connect to the mid-latitude circulation forming a large circular system, the Subtropical Gyre. The South Atlantic has a similar gyre in the subtropical region, but at higher latitudes we can notice a strong westerly current cutting all along the basin, essentially along a latitude line between 40S and 50S. Figure 1.6: The surface currents of the Atlantic ocean The North Pacific surface circulation is shown in Figure 1.7. We can notice here again a strong current along the Western boundary of the basin that than feed into a basin wide gyre that connects to the equatorial circulation. The boundary current, known her as the 18 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM Kuroshio Current, is very narrow and intense along the Japan coast, as it is also the case of the Gulf Stream in the Atlantic, and it tapers into a wide system of streams and eddies into the open ocean. It is possible to see also local system, like the small gyre off the Alaskan coast and similar circulation in the marginal seas, like the Sea of Okhotsk, near the Siberian coast. Their presence is remarkable as we are looking here at climatological averages over more than 40 years and so they are stable and persistent features. This is another reminder of how even relatively smaller feature in the ocean can be climatologically persistent over many years. Figure 1.7: The North Pacific surface circulation The circulation of the South Pacific Ocean is shown in Figure 1.8. The subtropical gyre is visible also here, but there is a weak indication of a western intensification current in the West pacific, close to the coasts of Australia and New Zealand. The strong high latitude current that we have seen in the Atlantic is also present here end evidently connect to the other basin through the Drake Passage, that is the Straits between South America and Antarctica. Figure 1.8: The circulation of the South Pacific Ocean The Indian Ocean circulation is shown in Figure 1.9. The Indian Ocean is confined by continental masses in the north, so it is mostly composed of the equatorial region and the midlatitudes are all in the Southern Hemisphere. The subtropical gyre is present, together with features North of Madagascar. A boundary current develops on the African coast, The westerly current in the southern mid-latitudes is visible also here, strong and with a vigorous eddy field. Figure 1.9: The Indian Ocean circulation 1.2. OCEANS 19 At this point we can suspect that there is probably a continuous ring of currents around Antarctica and this can be confirmed by the bottom panel of Figure 1.9 that shows the entire extent around a longitude circle of the current, known as the Antarctic Circumpolar Current. It is a strong, highly turbulent system that connects all the Ocean Basins. The equatorial circulation The Equator is a special place for the atmosphere and so it is a special place also for the Oceans. The circulation in this area is strongly coupled with the atmospheric circulation and it is often characterized by both westerly and easterly currents and by special behaviours right at the Equator line. Futhermore, it is also very different from ocean to ocean. The equatorial current system in the Pacific Ocean is shown in Figure 1.10. It is a complex system composed of two easterly currents, the North and South Equatorial currents, sand- wiching the westerly Equatorial countercurrent. It is possible to notice that at the Equator the currents are strongly easterly and diverging, leading to the emergence of upwelling at the Equator by Ekman transport. Figure 1.10: The equatorial current system in the Pacific Ocean The complexity of the equatorial current system can be further appreciated by looking at the vertical section of the zonal current at the Equator. A strong westerly current below the surface, slanting from the West Pacific to the East Pacific, is visible at depth between 200 and 300m. The speed is in excess of 1 m/s and it gets progressively shallower in the East. It is however part of an alternation of westerly and easterly currents that become progres- sively weaker as they get deeper. They are centered almost perfectly at the Equator, as it can be seen in Figure 1.11. The latitudinal position of the undercurrent is very tightly controlled by rotational effects and it precisely tracks the position of the equatorial line. Figure 1.11: Alternation of westerly and easterly currents The equatorial Atlantic Ocean surface circulation (Figure 1.14). It is possible to see the easterly South Equatorial Current that is straddling the Equator between 5S and 5N. The North equatorial Countercurrent is westerly and it covers the area between 5N and 10N, and the weaker North Equatorial Current is located at northern latitudes. The equatorial flow is divergent and also in this case we can presume the existence of upwelling at the Equator. A strong coastal current is visible as the Guinea Current, in the Gulf of Guinea. The South Equatorial Current feeds into the North Brazilian Current that then flows northward along the South America coast, taking different names as it finally emerges as the Caribbean Current in the Caribbean sea. A similar structure of alternating westerly and easterly undercurrents 20 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM Figure 1.12: The equatorial Atlantic Ocean surface circulation and the structure of currents and undercurrents exist also in the equatorial Atlantic (Figure 1.14), but it is weaker and only the first westerly maximum is well visible. It is also slanting towards the East, but the maximum is reached more towards the western boundary of the basin with respect to the Pacific Ocean. The deeper easterly jets are also much less weaker. The undercurrents jets are essentially absent in the Indian Ocean. The Gulf Stream The current system of the Gulf Stream is one of the major feature of the global ocean circulation. The Kuroshio The current system of the Kuroshio is one of the major features of the global ocean circulation. Figure 1.13: The Kuroshio on the left and Gulf Stream on the right Figure 1.14: The upwelling zones on the left and the Somali current on the right. 1.3. ICE 21 1.3 Ice Sea ice covers about 7% of Earth’s oceans and plays a critical role in the global climate sys- tem, acting as a barrier between the atmosphere and the ocean. It forms when seawater cools below its freezing point, creating ice floes that are often cracked and shifted by environmental forces. Despite its compact appearance, sea ice is dynamic and never fully covers open water areas due to constant movement and deformation. In the Southern Hemisphere, sea ice extends seasonally from about 3 million km2 in summer to 18 million km² in winter, covering around 7% of the Southern Ocean at its peak. The ice can reach as far as 60S during winter, particularly in the Weddell and Ross Seas. However, it tends to be less compact compared to the Arctic, largely due to divergent winds and the influence of ocean currents. By contrast, in the Northern Hemisphere, sea ice ranges from 7 million km2 in summer to 15 million km² in winter, making up about 10% of the ocean area during its maximum extent. The Arctic Ocean forms the core region of sea ice, with thick polar caps and thinner ice in surrounding seas like the Bering Sea and the Sea of Okhotsk. Ice can also drift into the North Atlantic through the Labrador and Greenland seas. The formation of sea ice depends on several factors. Salinity plays a crucial role, as higher salinity lowers the freezing point of seawater. In regions with salinity levels above 24.7%, convective mixing occurs as dense, salty water sinks, delaying the freezing process. Under calm condi- tions, supercooling can allow water to remain liquid below its freezing point until ice crystals form, which then accelerate the freezing of surrounding water. As the ice thickens, it insulates the ocean below, reducing heat loss and slowing further growth. Sea ice melts during summer, particularly in the Arctic, where increased sunlight and pro- longed daylight accelerate the process. Melting can occur at rates of about 40 mm per day, with melt ponds and drainage channels forming on the surface. The loss of ice is further aided by tidal forces and storms, leading to disintegration near coastlines and open water areas. By late summer, much of the peripheral sea ice in the Arctic disappears, leaving only the central ice pack. Sea ice properties vary greatly between the Arctic and Antarctic regions. In the Arctic, cen- tral ice thickness typically ranges from 2 to 4 meters, with thinner ice in peripheral seas (0.5 to 1 meter). Antarctic icebergs are far larger, often exceeding 600 meters in thickness. Ice density, generally between 880 and 910 kg/m3 , allows it to float. The topography of sea ice includes ridges and rubble zones formed by colliding ice sheets, with some ridges extending 10 to 15 meters below the surface. The movement of sea ice is also an important factor in ocean circulation and climate. In the Arctic, the Beaufort Gyre and the Transpolar Drift Stream transport ice across the Arctic Basin, eventually exiting through the Fram Strait into the North Atlantic. This ice contributes to cold currents like the East Greenland Current. In the Antarctic, ice drifts clockwise in the Weddell Sea, eventually melting in warmer waters and transporting freshwater to lower lati- tudes. Overall, sea ice is a vital component of Earth’s climate system. It influences heat exchange between the ocean and atmosphere, regulates salinity and density in ocean waters, and con- tributes to the broader dynamics of polar and global ocean circulation. Its seasonal cycles and properties make it an essential focus for understanding climate change and its impacts. 1.4 Interconnections between atmosphere, ocean and ice Important transfers of energy, mass, momentum occur between the oceans and the atmosphere and are greatly modified by the presence of sea ice. 22 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM 1.4.1 Atmosphere effect Wind forcing is a prime driver for upper ocean circulation and thereby impacts deep ocean circulation as well, and for sea ice thereby impacting its motions and location. In addition, air temperatures and moisture contribute to determining the energy fluxes across the interface between the atmosphere and surface, contributing to ice maintenance, growth and melt and to the temperature distribution in the ocean and water velocity. Fridtjorf Nanse noted that sea ice does not drift in the direction of the wind but at an angle of 20 − 40 to the right due to the Earth’s rotation and speculated that the motions in the water beneath sea ice deviate even more; the motion of each water layer deviate slower and farther to the right to the layer above and Ekman proved it mathematically adding that these influence stops at a depth of about 200 m. 1.4.2 Ocean effect Ocean has a significant impact on the other two components. It is essentially a limitless mois- ture source and supplies the majority of atmospheric water vapor. The ocean land contrast exerts a major control over the distribution of evaporation over the global precipitation. The thermal inertia of the ocean results in a much lesser seasonal temperature range at the open ocean surface than at a land surface or a sea ice surface. This creates a lesser atmospheric temperature range over open ocean than over land or sea ice, as cold winter air is warmed by the underlying ocean and warm summer air is cooled. Since the ocean also absorbs a signif- icant amount of the atmosphere’s carbon dioxide (CO2 ), it also helps to slow the buildup of atmospheric CO2. The oceans contribute significantly to the poleward transport of heat, by means of ocean currents. This transport influences the heat balances in both high and low latitudes and the average temperature gradient between the equator and the poles. The temperature distri- bution of the ocean surface is the prime determinant of the sea ice distribution, because, in general, ice forms where the water temperature has reached the freezing point. Similarly, the ocean salinity distribution is important to the formation of sea ice, because the salt content of the water affects the freezing temperature. Once ice is formed, warm currents entering an ice-covered region tend to melt the ice cover, and cold currents moving away from the ice region tend to carry ice with them. El Niño/Southern Oscillation (ENSO) This is an example of atmosphere/ocean interconnections one of the major large-scale pat- terns. The El Niño and Southern Oscillation phenomena were originally examined separately, before their close interconnection became apparent. El Niño originally referred to a seasonal warming of the waters along the coast of Peru, fre- quently occurring shortly after Christmas. The term is now generally restricted to large-scale warming events, extending well into the central equatorial Pacific. The episodes can last many months, and do not necessarily begin near Christmas. The Southern Oscillation, charac- terized by consistent sea level pressure, temperature, and precipitation changes in the South Pacific, was first discussed by Walker who found an alternating pressure pattern involving the normal southeast Pacific high pressure and the low pressure region near the Indian Ocean and western Pacific regions. Under normal conditions (non–El Niño), there is a strong sea surface temperature difference between the warm western Pacific and the cooler eastern Pacific, caused by upwelling along South America’s west coast driven by east-to-west trade winds. This setup creates a convec- tion cycle where air rises over the warm western Pacific, leading to heavy rainfall, and sinks over the cooler eastern Pacific, forming an east-west Walker circulation. 1.4. INTERCONNECTIONS BETWEEN ATMOSPHERE, OCEAN AND ICE 23 During El Niño, the trade winds weaken, reducing upwelling along the South American coast. This leads to warmer sea surface temperatures in the eastern Pacific, disrupting the tempera- ture gradient. As a result, air rising and rainfall patterns shift eastward, cooling the western Pacific and causing significant climate effects. For example, the Indonesian-Australian region may experience drought, while wetter conditions occur farther east. Figure 1.15: NAO North Atlantic Oscillation (NAO) The North Atlantic Oscillation (NAO) is a pressure oscillation measured by the sea level pressure difference between Iceland and the Azores, reflecting the strength of the Icelandic low-pressure system and the Azores high-pressure system. (although sometimes it is indexed instead by the pressure difference between Newfoundland and Lisbon, Portugal, or between Iceland and Lisbon). It has both short-term and long-term variations, significantly affecting North Atlantic climate patterns. Positive NAO phases strengthen the Icelandic low pressure, resulting in colder winters in eastern North America, warmer and wetter conditions in western Europe, reduced sea ice near Greenland and Scandinavia, and increased sea ice in Baffin Bay and the Labrador Sea. NAO impacts extend to global teleconnections, including influences on Russia, the Indian monsoon, and ocean temperatures. The positive NAO phase is associated with a tripolar sea surface temperature pattern: a cold anomaly in the subpolar North Atlantic, warm anomalies near Europe, and a cold subtropical anomaly near the Equator. The Gulf Stream further propagates these anomalies toward Europe, enabling atmospheric feedback and improving predictability of NAO patterns. The interaction between the ocean and the NAO remains a key area of research. 1.4.3 Ice The presence of sea ice has numerous climatic consequences, influencing the temperature and circulation patterns of both the atmosphere and the oceans. Sea ice lessens the amount of solar radiation absorbed at the ocean’s surface: only about 20˘50% of the incident solar ra- diation is absorbed, the rest being reflected to space and is therefore lost, while without the ice, typically 85˘95% is absorbed. It serves as a strong insulator, restricting exchanges of heat from 102 to 103 Wm−2 from the ocean to the atmosphere, to 10 to 20 Wm−2 ; mass, momentum, and chemical constituents between the ocean and atmosphere. In winter, ice cover enhances polar cooling by increasing temperature gradients between the poles and the equator, intensifying atmospheric circulation. However, stronger circulation brings warm air into polar regions, reducing these gradients and creating a negative feedback, making the overall effect on atmospheric circulation uncertain. In summer, ice acts as a ther- 24 CHAPTER 1. PHYSICAL DESCRIPTION OF THE CLIMATE SYSTEM mal insulator, reducing heat transfer from the atmosphere to the ocean. Its high albedo also reflects solar radiation, limiting heat absorption and increasing the net heat gain in polar oceans. These seasonal dynamics highlight the complex role of ice in the climate system. Another aspect of the insulation is the lessened evaporative transfer to the atmosphere in the presence of ice cover, resulting in reduction of moisture available for cloud formation, rain and snow. The freezing and melting of sea ice influence seasonal and regional climates by moderating temperature extremes. Ice formation releases heat, while melting absorbs heat, facilitating a net equatorward transport of heat and salt from polar regions. During ice formation, salt is rejected into the underlying ocean, increasing the salinity and density of the mixed ocean layer, which can lead to deep convection and the formation of bottom water that drives global ocean circulation. This process is most significant at the edges of ice packs, where winds create open water that freezes rapidly, enhancing local ice production. Approximately one-third of bottom water originates from ice formation along ice margins. These dynamics connect ice processes to global climate systems, influencing both local and large-scale ocean circulation. Chapter 2 Fundamental Equations and processes Last updated: 2024-12-06 Source file: chapter-fund-eqs-processes.tex 2.1 Fundamental Equations 2.1.1 Introduction Sample citation of Richardson, 1922. The atmosphere is in motion and it is a continuous mixing and clashing of vortices and structures, but when it is averaged over a long period of time (fig.2.1) it shows a remarkable simple structure. The figure shows the wind at an approximate level of about 12km, that is considered to be in the free atmosphere, far from the influence of the ground. The circulation is a large vortex around the pole that shows small oscillation in latitudes, especially pronounced over North America and the Asian Pacific Coast. The flow is therefore predominantly in the East-West direction, with a relatively small component in the meridional direction. The circumpolar vortex has been one of the first structures to be recognized when plentiful obser- vations of the upper air flow became available, but it provided some of the intriguing questions that drove the development of geophysical fluid dynamics to this time, some of them have not been completely understood. What is maintaining this peculiar circulation? Which factor determines the amplitude and location of the undulations in meridional direction? Some of these question will be addressed in these chapter. Figure 2.1: ERA5-wind 200 2.1.2 Coordinate systems Spherical Coordinates The most commonly used coordinate system for the analysis of the atmosphere and the oceans is a spherical coordinate system attached to the rotating Earth 25 26 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES (Fig. 2.2 ). The spherical coordinates are slightly different from the usual mathematical ones as the latitude is measured from the equator and therefore it can take negative values. The longitude is running west to east. The longitude is also known as the “zonal” direction whereas the latitude is also known as the "meridional" direction. Winds are identified by the direction they are coming from, so a "westerly" wind is coming from the West and an "easterly" wind is coming from the East. This coordinate system is rotating with the Earth and therefore it generates force terms in any dynamical equation expressed in this system of coordinate, the Coriolis terms. Figure 2.2: Coordinate system The Beta-plane It is sometimes convenient to shift coordinate system if the latitudinal extension of the motion is not too great with respect to the motion parameters as they are expressed in the adimen- sional numbers. When this is possible, a tangent coordinate system is applied at a specific latitude ϕ0 and the resultant Cartesian coordinates system is called the β-plane. Usually symbols (x, y) are used in this case for the zonal and meridional coordinate. In the β-plane the planetary vorticity f is linearized as f = f0 + βy, where β = ∂f ∂y (ϕ0 ). Advective derivative To describe the governing equation of the atmosphere and eventually of the ocean we have to understand how we write the rate of change with time of this fluid. This problem was solved by considering the fact that the rate of change in the fluid cannot be seen as rate of change with respect to a fixed system of coordinate because the system is moving with the fluid itself. Therefore, first we have to find a way to describe the change taking into account the moving system of restaurants. These can be done by using a concept developed in the 19th century by Euler that is called "advective derivative" that can be obtained from a total derivative of the property, dϕ ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂z ∂ϕ ∂ϕ = + + + = + v · ∇ϕ dt ∂t ∂t ∂x ∂t ∂y ∂t ∂z ∂t and so it can be defined as Dφ ∂φ = + v · ∇φ Dt ∂t 2.1. FUNDAMENTAL EQUATIONS 27 in this way the moving fluid can be described by derivatives with respect the "fixed" coordinate system, i.e. the Eulerian description. The alternative description of the observer moving with fluid is known as the "Lagrangian" description. 2.1.3 Equation of motion The motion of a physical system is governed by conservation laws: conservation of momentum, mass and energy. The conservation of momentum equation will identify the forces acting on the system. The conservation of energy will identify the processes capable of changing the energy of the system, i.e. thermodynamical processes. The equation governing the motion of the atmosphere can be written as: Du uv tan ϕ uw 1 ∂p − + =− + f v − fˆw + Fλ Dt r r ρr cos ϕ ∂λ Dv u2 tan ϕ vw 1 ∂p − + =− − f u + Fϕ Dt r r ρr ∂ϕ Dw u2 + v 2 1 ∂p − =− − g + fˆu + Fz Dt r ρ ∂z the f = 2Ω sin ϕ and fˆ = 2Ω cos ϕ terms arise from the rotating spherical coordinate system that we have chosen, other terms are generated by the spherical geometry. Some of them are small and traditionally they can be neglected, so that we arrive at the system   Du u tan ϕ 1 ∂p −v f + =− + Fλ Dt a ρa cos ϕ ∂λ   Dv u tan ϕ 1 ∂p +u f + =− + Fϕ Dt a ρa ∂ϕ Dw 1 ∂p =− − g + Fz Dt ρ ∂z where we have also used the Shallowness Approximation by assuming r = a+z ≈ a, where a is the Earth radius. However the advective derivative must be expressed in spherical cordinates D ∂ u ∂ v ∂ ∂ = + + +w Dt ∂t a cos ϕ ∂λ a ∂ϕ ∂z so that the velocity components are ∂λ u = a cos ϕ ∂t ∂ϕ v=a ∂t ∂z w= ∂t These equations govern the mechanical behaviour of the atmosphere, and we will see in a different form, also of the ocean. There three forces in action: pressure gradient, rotation via the Coriolis force and gravity. The equation are not complete,we have three equation but five variables, so we need to find the missing relations. We are using the basic conservation principles, the latter equations describe the conservation of momentum, we can exploit the conservation of mass. The mass of the fluid must be conserved locally, because there are now sinks or sources in the atmosphere itself, so we want to write the mass of a volume of atmosphere fixed in space as 28 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES Z M= ρ dV V the mass in the volume can only change if there is a flux of mass at surface S, Z Z ∂ ρ dV = − ρv · n dS ∂t V S using the divergence theorem however we have Z Z ∂ ρ dV = − ∇ · (ρv) dV ∂t V V because the volume is not changing with time we can bring the derivative inside the integral and we get Z ∂ρ + ∇ · (ρv) dV = 0 V ∂t but the volume is arbitrary, so it must be that ∂ρ + ∇ · (ρv) = 0 ∂t is valid locally. We have still at our disposal the conservation of thermodynamical energy and so we can also use the first law of thermodynamics for a gas, that is a statement of internal energy, where Cv is the specific heat for air at constant volume and T is the temperature in Kelvins.   DT D 1 cv = −p +Q Dt Dt ρ where we included the temperature and heating/cooling term Q (which is the net heat gain or loss to the external sources, for example the solar insolation, heating or cooling due to long wave radiation, latent heating due to condansation of water vapor into liquid water, and sensible heating due to conduction and convection). The state variable are then linked by the state equation p = ρRT where R is the gas constant for dry air. We can use the equation of state to write the energy equation (or the temperature equa- tion) in a different form,   DT D RT DT RT Dp cv = −p + Q = −R + +Q Dt Dt p Dt p Dt yielding the alternative forms ( since cp = cv + R), DT 1 Dp cp − =Q Dt ρ Dt For many purpose atmospheric and oceanic motions can be considered essentially adiabatic. However, for climate studies, the assumption of exclusively adiabatic processes is not appro- priate since the amount of heat added or lost to a unit volume of air or water over a long period of time can be substantial. For adiabatic processes Q = 0: 2.1. FUNDAMENTAL EQUATIONS 29 DT 1 Dp cp − =0 Dt ρ Dt cp DT R Dp − =0 T Dt p Dt D R D log T − log p = 0 Dt cp Dt integrating it we get  R/cp p log T /T0 − log = const p0 or  R/cp T p0 = const T0 p so the quantity, known as potential temperature  R/cp p0 θ=T p is conserved in adiabatic processes and the thermodynamics equation can be written as Dθ =Q Dt Hydrostatic balance. Under the action of gravity the vertical component of the pressure gradient force balances the action of gravity, resulting in very small vertical acceleration ∂p = −gρ ∂z then if we take the vertical derivative of the eq. Eq:logT  R/cp  R/cp −1 1 dT p0 p0 R T p0 dp − 2 =0 T0 dz p p cp T0 p dz simplifying  −1 dT p0 R p0 dp − 2 T =0 dz p cp p dz or dT 1 R dp dT 1R − T = + gρ T =0 dz p cp dz dz p cp but using the equation of state dT g =− dz cp that gives how the temperature change with height under adiabatic conditions and when the hydrostatic balance is valid. This is known as the adiabatic lapse rate. If I assume: −g Cp = 1000; = 10C/km Cp the temperature must drops linearly with height. 30 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES Figure 2.3: Geostrophic balance If we integrate over height z, assuming isothermal atmosphere, we get an exponential trend over density ρ and pressure p, which drops with height (because of the perfect gas law): −RT z p(z) = p0 e g Geostrophic balance. Geostrophic balance is the balance between Coriolis Force and the horizontal component of the pressure gradient force. The pressure gradient force (PGF) pushes air or water from high pressure to low pressure. The Coriolis force deflects the motion to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. In geostrophic balance, these forces are equal in magnitude and opposite in direction. The geostrophic balance assumes the flow is steady and does not accelerate, so it neglects the effects of inertia. This balance is most accurate for large-scale motions (e.g., planetary or synoptic scales) where friction and other forces are negligible. Vertical motions are typically much smaller than horizontal motions and are neglected in geostrophic balance. In geostrophic balance: Air or water flows parallel to the isobars (lines of constant pressure) or contours of constant geopotential height. The speed of the geostrophic flow increases with stronger pressure gradients (closer isobars). However, on small scales (e.g., tornadoes, boundary layers), friction and other forces become important, so geostrophic balance is less accurate. Near the Equator, The Coriolis parameter (f ) approaches zero, making the geostrophic balance invalid. Advective derivative in rotating coordinates. A vector in spherical coordinates is: u = îu + ĵv + k̂w but the unit vectors move in a rotating frame, so the advective derivative of a vector is given by: Du Du Dv Dw Dî Dĵ Dk̂ = î + ĵ + k̂ + u +v +w (2.1) Dt Dt Dt Dt Dt Dt Dt Coriolis Factors The Coriolis forces derives from the conservation of angular momentum. d (RA (ΩRA + u)) = 0 dt or dRA dRA du 2ΩRA +u + RA =0 dt dt dt rearraging du dRA = (2ΩRA + u) dt dt The Coriolis components then are 2Ω sin ϕ and 2Ω cos ϕ. For the horizontal and vertical components respectively: f = 2Ω sin ϕ. 2.1. FUNDAMENTAL EQUATIONS 31 Figure 2.4: Coriolis factors 2.1.4 Summary of fundamental equations- Summarizing our discussion, the fundamental equation that describe the motion of the atmo- sphere are considered in the following approximations: Hydrostatic approximation Shallow fluid approximation. The vertical coordinate r is sustituted by a + z(a >> z) except in differentiation Neglecting metric terms involving vertical velocity w The first approximation is independent, the second and the third must be applied together. These primitive (non hydrostatic) equations are:   Du u tan ϕ 1 1 ∂p −v f + =− + Fλ Dt a a cos ϕ ρ ∂λ   Dv u tan ϕ 1 1 ∂p +u f + =− + Fϕ Dt a a ρ ∂ϕ Dw 1 ∂p =− − g + Fz Dt ρ ∂z Dθ =Q Dt   ∂ρ 1 ∂ ∂ ∂ + (ρu) + (rv cos ϕ) + (ρw) = 0 ∂t a cos ϕ ∂λ ∂ϕ ∂z p = ρRT where we have used the divergence in spherical coordinates. These equations are still not closed because we will need to express the heating/cooling term Q and the friction terms F as a function of the state variables. This will require a theory of the processes that drive them. Where R = 287.052874J kg−1 K−1 is the gas constant for dry air and cp = 1.005 is the specific heat at constant pressure, cv = 0.718 is the specific heat at constant volume, κ = cRp and γ = cp /cv is their ratio. 2.1.5 Simplified equations- *domanda: Cri perchè abbiamo le eq. semplificate e poi riprendiamo quelle intere per parlare delle Nav.Sto.?* For theoretical and idealized studies the set of equation projected on the β-plane is also used. The β-plane approximation is a simplified model used in geophysical fluid dynamics to account the variation of the Coriolis parameter f with latitude, in this approximation f is linearized around a reference latitude such that f ≈ f0 + βy with y = a(ϕ − ϕ0 ) is the 32 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES northward displacement from the reference latitude. With the β-plane approximation, no rotation, no sphericity, neglecting the meridional coordinates , the equations become: Du 1 ∂p − fv = − + Fx Dt ρ ∂x Dv 1 ∂p + fu = − + Fy Dt ρ ∂y Dw 1 ∂p =− − g + Fz Dt ρ ∂z Dθ =Q Dt ∂ρ + ∇ · (ρv) = 0 ∂t p = ρRT and the gradient operator is the cartesian operator ∂ ∂ ∂ ∇= + + ∂x ∂y ∂z and the advective derivative is then D ∂ ∂ ∂ ∂ = +u +v +w Dt ∂t ∂x ∂y ∂z The first set of fundamental equations that we encounter is the conservation of Momentum (Navier-Stokes equations). The Navier-Stokes equations describe the motion of fluid substances like air and water. They account for forces due to pressure, viscosity, and external forces such as gravity. This set of equations is key to understanding how winds, ocean currents, and other flows evolve due to internal and external forces. In a rotating frame of reference, specifically describing the motion of a fluid (e.g. the atmosphere or ocean currents) the principal equation can be declined in the different directions: longitude, latitude and vertical directions. The left-hand side of the equations begins with material derivative of the velocity compo- nents (u in the east-west direction, associated with longitude; v north-south velocity compo- nent, associated with latitude, w is the vertical velocity component). The material derivative includes both the local rate of change and the advection (transport) of the velocity. This term tells us how the velocity of a fluid parcel changes over time, taking into account both temporal changes and the movement of the parcel. The second term on the left represents the Coriolis force and the centrifugal force in the rotating reference frame of the Earth. f is the Coriolis parameter, which depends on the latitude ϕ and is given by f = 2Ω sin ϕ where Ω is the angular velocity of the Earth and ϕ is the latitude. The Coriolis force is proportional to the velocity and acts perpendicular to the motion of the fluid, deflecting the fluid in different directions depending on the hemisphere. On the right side of the equations we find the pressure gradient force in the longitudinal (first) and latitudinal (second) directions, which drives fluid motion due to differences in pres- sure. The terms Fλ , Fϕ and Fz represent an additional force term that could represent any other forces acting on the fluid parcel in the longitudinal, latitudinal and vertical direction. It might include friction, external forces, or any other model-specific forces not accounted for in the other terms. Let’s now focus on them one by one.   Du u tan ϕ 1 1 ∂p −v f + =− + Fλ (2.2) Dt a a cos ϕ ρ ∂λ 2.1. FUNDAMENTAL EQUATIONS 33 The term u tana ϕ accounts for the centrifugal force resulting from the Earth’s rotation. Here, u is the east-west velocity, ϕ is the latitude, and a is the Earth’s radius. The centrifugal force ∂p is stronger near the equator, and this term adjusts the Coriolis effect to account for that. ∂λ is the derivative of pressure with respect to longitude (λ), indicating how pressure changes as you move east or west. The term a cos ϕ accounts for the spherical geometry of the Earth and the fact that distances between lines of longitude vary with latitude (they are widest at the Equator and shrink towards the poles). This term describes the acceleration due to the horizontal pressure gradient in the longitudinal direction, with the pressure gradient force causing flow from regions of higher to lower pressure.   Dv v tan ϕ 1 1 ∂p +u f + =− + Fϕ (2.3) Dt a a ρ ∂ϕ The term v tana ϕ is the centrifugal force term in the north-south direction, considering Earth’s p curvature. ϕ represents the pressure gradient in the latitude direction, and it drives motion from high to low pressure. Note that the Coriolis term arises due to the rotation of the Earth, which introduces an apparent force in a rotating reference frame. This term depends explicitly on the latitude because of how the Earth’s rotation affects the direction and magnitude of the Coriolis effect at different points on the Earth’s surface. The Coriolis force explicitly depends on the sine of the latitude, as the projection of Ω onto the local horizontal plane is Ω sin ϕ (the latitude ϕ determines the angle between the Earth’s rotational axis and the local vertical. Ω is the angular velocity vector of Earth’s rotation, it points along the axis of rotation (towards the North Pole). At the Equator ϕ = 0 the Coriolis effect is perpendicular to Ω and has a max horizontal effect. At the poles (ϕ = ±90) the effect aligns with Ω, and only the vertical motion is affected. In the vertical direction: Dw 1 ∂p =− − g + Fz (2.4) Dt ρ ∂z While the Coriolis force primarily affects horizontal motion, the vertical component of the Coriolis force is negligible because Ω is nearly parallel to the vertical axis in most regions. In rotating systems like the atmosphere or oceans, vertical Coriolis terms are often ignored. g acts downward and opposes upward motion. The equation Dθ =Q (2.5) Dt says that the rate of change of θ (the material derivative) experienced by a fluid par- cel moving through space is equal to the source term Q. This equation describes the rate of change of a scalar quantity (like temperature or moisture) for a fluid parcel moving through space. The term Q could represent a source of energy or mass, such as heat from the sun or moisture added by evaporation. In the context of a weather or climate model, this would represent how temperature (or another variable) changes as the air parcel moves and as energy is gained or lost. This equation is a general form for describing the time evolution of a scalar quantity (like T or moisture) in a moving fluid, where the change in that quantity is driven by external sources or sinks. The conservation of mass or continuity equation ensures that the mass is conserved in the system. For the atmosphere or the ocean, it expresses how the density of air or water changes over time due to processes like flow and diffusion. ∂ρ + ∇ · (ρv) = 0 (2.6) ∂t 34 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES ∇ · (ρv) represents the divergence of the mass flux, which accounts for the movement of the mass. The last one is the equation of state for atmospheric or oceanic models links pressure, density and temperature of the fluid. The ideal gas law is typically used: p = ρRT (2.7) 2.1.6 Linear solutions We look for solution that describe small oscillation away from a basic state, in this case assumed in hydrostatic balance with constant wind of velocity u, and I’m looking at deviation from this basic state: ∂p0 gp0 = −gρ0 = − (2.8) ∂z RT0 With the further assumption of isotherm atmosphere we can integrate to get: p0 (z) = pR e−z/H (2.9) and H = RT0 g is the scale height. As we have seen, the primitive equations are a system of nonlinear partial differential equations governing fluid motion on a rotating sphere. These include momentum equations, continuity (mass conservation), thermodynamic equations and the equation of state. Since solving the full nonlinear system is often impractical, the system is linearized to analyze small deviations (or perturbations) from a reference state (steady flow or a state of rest). Assume the flow can be separated into a basic state and a perturbation: u = u0 + u′ v = v0 + v ′ θ = θ0 + θ ′ p = p0 + p′ The resulting linearized primitive equations describe how perturbations evolve over time and space (linear equations are obtained for the prime variables neglecting all terms quadratic in perturbation). For instance, the pressure gradient becomes: 1 1 1 1 1 ′ ∇(p0 + p′ ) + g k̂ ≈ ′ ∇p′ + g k̂ = ρ ′ ′ ∇p + g k̂ = ∇p′ + g k̂ (2.10) ρ0 + ρ ρ0 + ρ ρ0 1 + ρ0 ρ0 and the potential temperature: k pkR p1−k 1 p′ ρ′      pR θ=T = ≈ θ0 1 + − p R ρ γ p0 ρ0 hence, θ′ 1 p′ ρ′ = − (2.11) θ0 γ p0 ρ0 cp with γ = cv , k= cv , R R = cp − cv. 2.1.7 Waves The linearized system often supports wave-like solutions. These solutions arise naturally due to the restoring forces in the equations, such as: 2.2. HOMOGENEOUS FLOWS 35 Pressure gradient drives the oscillations through compressibility or buoyancy Coriolis force introduces rotational effects leading to inertial and planetary waves Gravity acts as a restoring force for vertical displacement, driving internal gravity waves. We take the equations for perturbation: ∂u ∂u 1 ∂p +U + =0 ∂t ∂x ρ0 ∂x ∂w ∂w 1 ∂p ρ +U + +g =0 ∂t ∂x ρ0 ∂z ρ0 ∂ρ ∂ρ ∂ρ0 ∂u ∂w +U +w + ρ0 ( + )=0 ∂t ∂x ∂z ∂x ∂z ∂θ ∂θ ∂θ0 +U +w =0 ∂t ∂x ∂z θ ρ 1 p + + =0 θ0 ρ0 γ p0 Basically to find solutions, assume the perturbations take the form of plane waves: u′ (x, t) = ûei(k·x−ωt) where û is the amplitude of the perturbation, k is the wavevector, ω the angular velocity. Substituing this kind of wave into the linearized equations leads to a dispersion relation, which relates ω to k and the physical parameters of the system (Coriolis,...). The perturbation equations link the restoring forces to wave properties: 1. The momentum equations describe how velocity perturbations interact with pressure gradients and Coriolis forces. 2. The continuity equation ensures mass conservation, connecting velocity and density perturbations. 3. The thermodynamic equation relates temperature, pressure and density perturbations Imposing the hydrostatic approximation from the beginning would mean putting to zero from the start the time derivative of w in the momentum equation in the vertical. This basically is equivalent to say that we are removing the explicit dependence of the vertical velocity on time and therefore we will get waves similar to the Boussinesq approximation. The hydrostatic approximation eliminates vertical propagating sound waves, but the Lamb waves still exists. 2.2 Homogeneous flows Figure 2.5: Homogeneous flow The motion of the atmosphere is most appropriately described by three-dimensional equa- tions that describes the horizontal and vertical motion of the fluid, however a lot can be 36 CHAPTER 2. FUNDAMENTAL EQUATIONS AND PROCESSES understood by considering a simpler system that consider the motion of a free surface of a inviscid, homogeneous and incompressible fluid. These equations are variously referred to as one-level primitive equations or the shallow water equations. The system is described in Fig. 2.5, it is an homogenous layer of fluid covering the entire spherical planet. We have included a bottom topography h(x, y) and a deformation of the free surface η(x, y). The convention is such positive deformation of the surface are increasing the depth of the fluid, whereas positive deformation of the bottom are decreasing it. We have introduced here cartesian coordinates (x, y) such that dx ≈ a cos θdλ and dy ≈ adθ, we will also define f = 2Ω sin θ, then the incompressible equations of motion can be written as ∂u ∂u ∂u 1 ∂p = −u −v + fv − ∂t ∂x ∂y ρ ∂x ∂v ∂v ∂v 1 ∂p = −u −v − fu − ∂t ∂x ∂y ρ ∂y ∂w ∂w ∂w 1 ∂p = −u −v +g− ∂t ∂x ∂y ρ ∂z ∂u ∂v ∂w + + =0 ∂x ∂y ∂z If the aspect ratio δ ≈ H/L is small then hydrostatic balance is maintained to order δ 2 and so ∂p = −ρg + O(δ 2 ) ∂z because the density is constant we can integrate