Between Shemini Atzeret and Pesach, “Mashiv Haruah Umorid Hageshem”, which mentions Hashem’s might in causing the rain to fall, is recited in the second blessing of the Amida. Between Pesach and Shemini Atzeret, “Morid Hatal” is recited in its place, and it praises Hashem for creating dew. It should be noted that unlike the ninth blessing of the Amida (Barechenu/Barech Alenu), in which one requests rain or dew, Morid Hatal And Mashiv Haruah are not requests but rather praises.
There are instances in which rainfall during the summer months is considered detrimental or even a curse, and therefore, the Shulhan Aruch (Orah Haim, 114:4) says that if one accidentally recited “Mashiv Haruah” between Pesach and Shemini Atzeret, one would have go back to the beginning of the second blessing of the Amida. If one already recited the blessing of “Mehaye Hametim” and had accidentally said “Mashiv Haruah” in the summer months, one would have to go to the beginning of the Amida and start over. Furthermore, the Shulhan Aruch says that even if “Mashiv Haruah” was recited in a locality where rain is actually beneficial in the summer months, one would still need to go back.
A question arises regarding the recital of Mashiv Haruah in the southern hemisphere. Although rain may be detrimental in the summer, the period between Pesach and Shemini Atzeret actually coincides with their winter, and thus the status of Mashiv Haruah may be unclear there. Rabbi Haim Shabtai (Shu’t Maharhash/Torat Haim), answering questions from the early Jewish settlers in Brazil in the sixteenth century, entertains the possibility that Mashiv Haruah is not related to a specific part of the world, but rather to the unique needs of a country. Rabbi Mordechai Lebhar (Birkat Erev, § 8) suggests that there is a distinction between Mashiv Haruah and the blessing of Barech Alenu. Mashiv Haruah is a general mention of praise for how Hashem runs the world, and since most people live in the northern hemisphere, then it is appropriate to mention the praise that befits the season inthat part of the world, Therefore, even if one is in Australia during their summer, for example, one should still recite Mashiv Haruah.
On the other hand Barechenu and Barech Alenu are personal requests for dew and rain, respectively, and thus this is more location-dependent. Indeed, Rabbi Shmuel Wosner (Shevet Halevi, vol. I, § 21) suggests that people living in the southern hemisphere should recite Barechenu in their summer. Nevertheless, since there is doubt in the matter, he recommends that one insert the words “Veten Tal Umatar” (lit. “send dew and rain”), which is normally said in winter, in the blessing of Shome’a Tefila. Similarly, Rabbi Ben Zion Abba Shaul (Or Lezion, vol. II, ch. 7) says that in such a case, one should say “Veten Tal Umatar Be’artzenu Hakedosha” (lit. “send dew and rain in our holy land”) in Shome’a Tefila to reflect that the main thrust of these mentions of praise and requests is to pray for proper weather in the Land of Israel. Furthermore, in their winter, they should recite Barechenu since it is summer in the northern hemisphere, but they should still recite “Veten Tal Umatar” in Shome’a Tefila to request rain for their own land.
Rabbi Yosef Benaim (Noheg Behochma, pg. 225) writes about Rabbi Yisachar Bensoussan (Ibur Shanim), who witnessed in his native Fes the rabbis gathering in the city square during a drought, reading different parts from the Torah and Nevi’im and holding thirteen fasts, just as is explained in the Gemara (Ta’anit). He writes that after they did this, it began to rain and they read Hallel in gratitude of the Hashem answering their prayers.
Summary: In the southern hemisphere, one should recite Morid Hatal and Barech Alenu in their winter. In their summer one should recite Mashiv Haruah and Barechenu, and add Veten Tal Umatar Be’artzenu Hakedosha in the blessing of Shome’a Tefila.